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            Lecture 15: Linear Recurrences
         
    
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<p><strong>Description:</strong> Covers the mechanics of solving general linear recurrences as well as applications to the graduate student job problem and Fibonacci modeling of populations.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<p><span m="390">The</span> <span m="520">following</span> <span m="950">content</span> <span m="1550">is</span> <span m="1660">provided</span> <span m="2110">under</span> <span m="2390">a</span> <span m="2430">Creative</span> <span m="2830">Commons</span> <span m="3230">license.</span> <span m="4340">Your</span> <span m="4530">support</span> <span m="5040">will</span> <span m="5200">help</span> <span m="5440">MIT</span> <span m="5890">OpenCourseWare</span> <span m="6680">continue</span> <span m="7190">to</span> <span m="7270">offer</span> <span m="7690">high-quality</span> <span m="8440">educational</span> <span m="9070">resources</span> <span m="9700">for</span> <span m="9850">free.</span> <span m="11050">To</span> <span m="11060">make</span> <span m="11260">a</span> <span m="11310">donation</span> <span m="12000">or</span> <span m="12260">view</span> <span m="12700">additional</span> <span m="13130">materials</span> <span m="13670">from</span> <span m="13810">hundreds</span> <span m="14250">of</span> <span m="14360">MIT</span> <span m="14780">courses,</span> <span m="15900">visit</span> <span m="16110">MIT</span> <span m="16530">OpenCourseWare</span> <span m="17580">at</span> <span m="17740">ocw.mit.edu.</span> </p>
<p><span m="23340">PROFESSOR: All</span> <span m="23420">right.</span> <span m="23620">So</span> <span m="23770">last</span> <span m="24120">time</span> <span m="24350">we</span> <span m="24470">talked</span> <span m="24930">about</span> <span m="25570">methods</span> <span m="25960">for</span> <span m="26070">solving</span> <span m="26490">recurrences,</span> <span m="27490">and</span> <span m="27700">we</span> <span m="27790">spent</span> <span m="28040">most</span> <span m="28340">of</span> <span m="28420">our</span> <span m="28530">time</span> <span m="28820">talking</span> <span m="29190">about</span> <span m="29510">divide-and-conquer</span> <span m="30480">recurrences.</span> <span m="31630">These</span> <span m="31880">are</span> <span m="32009">recurrences</span> <span m="32560">where</span> <span m="32659">you</span> <span m="32750">break</span> <span m="33050">the</span> <span m="33140">problem</span> <span m="33550">up</span> <span m="33790">into</span> <span m="34470">much</span> <span m="34730">smaller</span> <span m="35330">sub-problems,</span> <span m="36040">like</span> <span m="36200">half</span> <span m="36530">the</span> <span m="36620">size</span> <span m="37210">or</span> <span m="37700">2/3</span> <span m="38230">the</span> <span m="38330">size.</span> <span m="39570">And</span> <span m="39840">they</span> <span m="39960">come</span> <span m="40130">up</span> <span m="40330">quite</span> <span m="40600">a</span> <span m="40640">bit</span> <span m="40800">in</span> <span m="40870">computer</span> <span m="41240">science</span> <span m="41690">when</span> <span m="41780">you're</span> <span m="41850">doing</span> <span m="42080">algorithm</span> <span m="42740">design</span> <span m="43190">and</span> <span m="43290">algorithm</span> <span m="43690">analysis.</span> </p>
<p><span m="45150">Today,</span> <span m="45620">we're</span> <span m="45790">going</span> <span m="45900">to</span> <span m="45990">spend</span> <span m="46280">our</span> <span m="46380">time</span> <span m="46720">talking</span> <span m="47060">about</span> <span m="47240">a</span> <span m="47300">different</span> <span m="47670">kind</span> <span m="47880">of</span> <span m="47960">recurrence</span> <span m="48690">that's</span> <span m="48880">called</span> <span m="49230">a</span> <span m="49300">linear</span> <span m="49670">recurrence.</span> <span m="50770">They</span> <span m="50970">also</span> <span m="51300">come</span> <span m="51470">up</span> <span m="51590">in</span> <span m="51690">computer</span> <span m="52060">science</span> <span m="52610">and</span> <span m="52870">a</span> <span m="52930">lot</span> <span m="53240">of</span> <span m="53330">other</span> <span m="53520">fields.</span> <span m="55360">Now,</span> <span m="55530">I'm</span> <span m="55650">going</span> <span m="55760">to</span> <span m="55800">give</span> <span m="55990">you</span> <span m="56070">the</span> <span m="56190">formal</span> <span m="56740">definition</span> <span m="57390">of</span> <span m="57480">a</span> <span m="57540">linear</span> <span m="57920">recurrence</span> <span m="58700">later.</span> <span m="59650">First</span> <span m="60030">I</span> <span m="60070">want</span> <span m="60270">to</span> <span m="60330">start</span> <span m="60870">with</span> <span m="61080">an</span> <span m="61150">example,</span> <span m="62990">and</span> <span m="63190">this</span> <span m="63350">is</span> <span m="63450">an</span> <span m="63530">example</span> <span m="63950">of</span> <span m="64019">a</span> <span m="64080">linear</span> <span m="64430">recurrence</span> <span m="64940">that</span> <span m="65000">comes</span> <span m="65319">up</span> <span m="65480">in</span> <span m="65590">population</span> <span m="66290">modeling.</span> <span m="66800">In</span> <span m="67070">fact,</span> <span m="67220">it</span> <span m="67340">comes</span> <span m="67570">up</span> <span m="67680">in</span> <span m="68005">a</span> <span m="68260">lot</span> <span m="68440">of</span> <span m="68510">places.</span> </p>
<p><span m="70260">And</span> <span m="70470">to</span> <span m="70620">start it</span> <span m="70970">off,</span> <span m="71200">we're</span> <span m="71330">going</span> <span m="71440">to</span> <span m="71650">analyze</span> <span m="72440">a</span> <span m="72490">particular</span> <span m="73030">problem</span> <span m="73460">that</span> <span m="73580">we</span> <span m="73740">call</span> <span m="74080">the</span> <span m="75300">graduate</span> <span m="75850">student</span> <span m="76350">job</span> <span m="76760">problem</span> <span m="77860">or</span> <span m="78040">the</span> <span m="78250">graduate</span> <span m="78710">student</span> <span m="79040">job</span> <span m="79390">nightmare.</span> <span m="81810">And</span> <span m="83000">the</span> <span m="83110">question</span> <span m="83570">here</span> <span m="84090">is,</span> <span m="84520">will</span> <span m="84700">your</span> <span m="84890">TAs</span> <span m="86150">be</span> <span m="86290">able</span> <span m="86450">to</span> <span m="86520">get</span> <span m="86670">a</span> <span m="86720">job</span> <span m="87150">as</span> <span m="87300">a</span> <span m="87420">professor</span> <span m="87950">somewhere</span> <span m="88350">when</span> <span m="88480">they</span> <span m="88790">get</span> <span m="88950">their</span> <span m="89080">PhD?</span> <span m="96100">So</span> <span m="96260">this</span> <span m="96450">is</span> <span m="96550">a</span> <span m="96620">problem</span> <span m="96990">they</span> <span m="97160">worry</span> <span m="97490">about</span> <span m="97740">a</span> <span m="97800">lot.</span> </p>
<p><span m="99040">And</span> <span m="99240">the</span> <span m="99380">idea</span> <span m="99820">in</span> <span m="99950">this</span> <span m="100130">problem</span> <span m="100620">is</span> <span m="100860">that</span> <span m="101440">there's</span> <span m="101650">some</span> <span m="102540">discipline,</span> <span m="103280">say</span> <span m="103550">computer</span> <span m="104000">science.</span> <span m="105270">And</span> <span m="106060">if</span> <span m="106210">you</span> <span m="106320">look</span> <span m="106470">in</span> <span m="106520">all</span> <span m="106780">the</span> <span m="106900">universities</span> <span m="107490">in</span> <span m="107540">the</span> <span m="107640">world,</span> <span m="108310">there's</span> <span m="109190">M</span> <span m="110010">faculty</span> <span m="110830">lines.</span> <span m="111640">So</span> <span m="111790">the</span> <span m="111930">total</span> <span m="112270">number</span> <span m="112520">of</span> <span m="112610">jobs</span> <span m="113240">is</span> <span m="113440">M,</span> <span m="120820">and</span> <span m="120980">that's</span> <span m="121170">a</span> <span m="121260">fixed</span> <span m="121740">value.</span> <span m="122790">Because</span> <span m="123120">of</span> <span m="123180">budgetary</span> <span m="123910">constraints,</span> <span m="124690">universities</span> <span m="125410">aren't</span> <span m="125590">growing,</span> <span m="126680">and</span> <span m="126910">so</span> <span m="127010">they're</span> <span m="127110">not</span> <span m="127330">going</span> <span m="127450">to</span> <span m="127660">create</span> <span m="128050">more</span> <span m="128530">computer</span> <span m="128960">science</span> <span m="129430">faculty</span> <span m="129840">positions.</span> <span m="130680">It's</span> <span m="130870">going</span> <span m="130990">to</span> <span m="131120">stay</span> <span m="131440">fixed</span> <span m="132050">over</span> <span m="132310">time.</span> <span m="139900">So</span> <span m="140130">it's</span> <span m="140260">a</span> <span m="140320">constant.</span> </p>
<p><span m="142990">And</span> <span m="144640">every</span> <span m="145100">year,</span> <span m="146950">professors</span> <span m="147810">generate</span> <span m="148360">more</span> <span m="148970">graduates</span> <span m="149610">who</span> <span m="149720">become</span> <span m="150260">professors.</span> <span m="151680">And</span> <span m="151830">in</span> <span m="151900">particular,</span> <span m="152780">we're</span> <span m="153090">going</span> <span m="153320">to</span> <span m="153400">assume</span> <span m="154030">that</span> <span m="154580">in</span> <span m="154730">this</span> <span m="154970">field</span> <span m="155830">every</span> <span m="156150">professor</span> <span m="156680">graduates</span> <span m="157390">1</span> <span m="157940">student</span> <span m="159000">who</span> <span m="159130">goes</span> <span m="159400">on</span> <span m="159610">to</span> <span m="159670">become</span> <span m="160000">a</span> <span m="160060">professor,</span> <span m="160770">or</span> <span m="161230">tries</span> <span m="161570">to</span> <span m="161670">if</span> <span m="161780">there's</span> <span m="161960">jobs.</span> <span m="163650">So</span> <span m="163890">each</span> <span m="164120">professor</span> <span m="168070">generates</span> <span m="172180">1</span> <span m="173290">graduate</span> <span m="174910">who</span> <span m="175000">becomes</span> <span m="175370">a</span> <span m="175440">new</span> <span m="175650">professor,</span> <span m="176840">as</span> <span m="176940">long</span> <span m="177170">as</span> <span m="177260">there's</span> <span m="177500">jobs,</span> <span m="180070">per</span> <span m="180310">year.</span> </p>
<p><span m="183460">With</span> <span m="183680">one</span> <span m="183910">exception,</span> <span m="184770">and</span> <span m="184930">that's</span> <span m="185200">first-year</span> <span m="185990">professors</span> <span m="187280">because</span> <span m="187680">they're</span> <span m="187940">too</span> <span m="188150">busy</span> <span m="188630">learning</span> <span m="189000">how</span> <span m="189080">to</span> <span m="189200">teach,</span> <span m="189760">getting</span> <span m="190030">grants,</span> <span m="190890">doing</span> <span m="191190">administrative</span> <span m="191840">stuff,</span> <span m="192540">just</span> <span m="192800">figuring</span> <span m="193220">out</span> <span m="193400">how</span> <span m="193620">it</span> <span m="193670">all</span> <span m="193920">works.</span> <span m="194710">So</span> <span m="194830">they</span> <span m="194940">don't</span> <span m="195100">have</span> <span m="195290">time</span> <span m="195620">to</span> <span m="195730">produce</span> <span m="196170">any</span> <span m="196340">grad</span> <span m="196630">students.</span> <span m="203510">So</span> <span m="203630">first</span> <span m="204040">year</span> <span m="204420">or</span> <span m="204530">new</span> <span m="204780">professors</span> <span m="206810">don't</span> <span m="207040">produce</span> <span m="207390">anything.</span> <span m="213340">Except</span> <span m="213410">first-year</span> <span m="213950">profs--</span> <span m="215130">oops,</span> <span m="216080">let</span> <span m="216190">me</span> <span m="216310">correct</span> <span m="216630">that.</span> <span m="223280">So</span> <span m="223330">first-year</span> <span m="223760">profs</span> <span m="224200">have</span> <span m="224350">0.</span> </p>
<p><span m="233340">Now,</span> <span m="234240">matters</span> <span m="234670">are</span> <span m="234730">made</span> <span m="235010">worse</span> <span m="235530">by</span> <span m="235670">the</span> <span m="235790">fact</span> <span m="236130">that</span> <span m="236240">Congress</span> <span m="236790">passed</span> <span m="237180">a</span> <span m="237240">law,</span> <span m="237750">and</span> <span m="237880">this</span> <span m="238050">is</span> <span m="238150">true,</span> <span m="239280">that</span> <span m="239470">more</span> <span m="239680">or</span> <span m="239700">less</span> <span m="239950">bars</span> <span m="240510">mandatory</span> <span m="241240">retirements</span> <span m="242460">in</span> <span m="242770">colleges.</span> <span m="244730">And</span> <span m="244890">so</span> <span m="244990">that</span> <span m="245200">means</span> <span m="245660">there</span> <span m="245900">are</span> <span m="246120">no</span> <span m="246370">retirements</span> <span m="247150">in</span> <span m="247250">this</span> <span m="247390">problem,</span> <span m="249780">and</span> <span m="250020">we're</span> <span m="250140">going</span> <span m="250250">to</span> <span m="250310">assume</span> <span m="250720">the</span> <span m="250800">faculty</span> <span m="251290">live</span> <span m="251530">forever.</span> <span m="254150">And</span> <span m="254400">so</span> <span m="254580">once</span> <span m="254810">you</span> <span m="254910">fill</span> <span m="255120">a</span> <span m="255170">position,</span> <span m="255620">it's</span> <span m="255810">filled</span> <span m="256310">forever.</span> <span m="256870">It</span> <span m="256950">doesn't</span> <span m="257300">exist</span> <span m="257670">anymore.</span> <span m="258160">In</span> <span m="258260">fact,</span> <span m="258800">if</span> <span m="258850">you</span> <span m="258950">walk</span> <span m="259180">around</span> <span m="259410">the</span> <span m="259480">math</span> <span m="259790">department,</span> <span m="261149">you</span> <span m="261529">can</span> <span m="261660">see</span> <span m="261790">the</span> <span m="261959">impact</span> <span m="262390">of</span> <span m="262470">this.</span> </p>
<p><span m="263160">[LAUGHTER]</span> </p>
<p><span m="263650">I</span> <span m="264990">think</span> <span m="265230">the</span> <span m="265360">median</span> <span m="265820">age</span> <span m="266180">is</span> <span m="266310">now</span> <span m="266560">well</span> <span m="267110">into</span> <span m="267330">the</span> <span m="267450">70s.</span> <span m="269390">And</span> <span m="269570">there</span> <span m="269660">will</span> <span m="269850">be</span> <span m="269970">actually a</span> <span m="270280">phenomenon</span> <span m="270950">over</span> <span m="271100">the</span> <span m="271210">next</span> <span m="271470">10</span> <span m="271680">years</span> <span m="272570">as</span> <span m="272800">the</span> <span m="272880">math</span> <span m="273130">faculty</span> <span m="273510">progress</span> <span m="273890">into</span> <span m="274060">their</span> <span m="274210">80s</span> <span m="275150">where</span> <span m="275410">they</span> <span m="275540">actually, in</span> <span m="276020">reality,</span> <span m="276380">probably</span> <span m="276710">do</span> <span m="276920">start</span> <span m="277190">to</span> <span m="277280">retire.</span> <span m="278170">And</span> <span m="278310">there will</span> <span m="278370">be</span> <span m="278480">a</span> <span m="278570">whole</span> <span m="278920">new</span> <span m="279120">wave</span> <span m="279460">of</span> <span m="279570">people</span> <span m="279870">hired</span> <span m="280170">in</span> <span m="280240">mathematics,</span> <span m="281280">just as</span> <span m="281740">happened</span> <span m="282180">back</span> <span m="282460">in</span> <span m="282530">the</span> <span m="282620">'50s</span> <span m="283030">and</span> <span m="283150">'60s</span> <span m="284310">after</span> <span m="284520">the</span> <span m="284780">Sputnik</span> <span m="285110">crisis</span> <span m="285590">where</span> <span m="285720">a</span> <span m="285750">lot</span> <span m="285950">of</span> <span m="286010">mathematicians</span> <span m="286630">were</span> <span m="286750">hired.</span> <span m="287810">And</span> <span m="287880">then</span> <span m="287960">they</span> <span m="288070">stayed</span> <span m="288370">in</span> <span m="288450">those</span> <span m="288640">positions,</span> <span m="289970">for</span> <span m="290120">our</span> <span m="290230">purposes,</span> <span m="290700">forever.</span> </p>
<p><span m="292530">Now,</span> <span m="293180">the</span> <span m="293330">question</span> <span m="293780">is,</span> <span m="294040">when</span> <span m="294640">do</span> <span m="294820">all</span> <span m="295170">the</span> <span m="295270">jobs</span> <span m="295770">get</span> <span m="295960">filled?</span> <span m="304640">So</span> <span m="304850">when are</span> <span m="305180">all</span> <span m="305540">M</span> <span m="305893">jobs</span> <span m="307700">filled</span> <span m="308350">by</span> <span m="308470">this</span> <span m="308680">process?</span> <span m="312830">Now,</span> <span m="313140">to</span> <span m="313320">be</span> <span m="313440">able</span> <span m="313560">to answer</span> <span m="313930">this</span> <span m="314150">question,</span> <span m="314600">we</span> <span m="314690">need</span> <span m="314890">one</span> <span m="315300">more</span> <span m="315560">piece</span> <span m="315860">of</span> <span m="315950">information.</span> <span m="318790">Can</span> <span m="318920">anybody</span> <span m="319340">think</span> <span m="319670">about</span> <span m="320000">one</span> <span m="320380">more</span> <span m="320890">fact</span> <span m="321300">here</span> <span m="321460">that</span> <span m="321570">we</span> <span m="321690">need</span> <span m="322050">before</span> <span m="322310">we</span> <span m="322410">start</span> <span m="322820">going</span> <span m="323040">off</span> <span m="323240">and</span> <span m="323360">answering</span> <span m="323670">this</span> <span m="323860">question,</span> <span m="324770">analyzing</span> <span m="325055">it?</span> <span m="326380">Yeah?</span> </p>
<p><span m="326880">AUDIENCE: How many professors</span> <span m="327305">do we</span> <span m="327730">start</span> <span m="328155">with?</span> </p>
<p><span m="328580">PROFESSOR: How</span> <span m="328880">many</span> <span m="329100">professors</span> <span m="329660">do</span> <span m="329770">we</span> <span m="329880">start</span> <span m="330270">with?</span> <span m="330480">What's</span> <span m="330740">the</span> <span m="331070">boundary</span> <span m="331540">condition?</span> <span m="332320">What's</span> <span m="332640">the</span> <span m="332740">base</span> <span m="333140">case</span> <span m="333460">if</span> <span m="333570">we</span> <span m="333650">were</span> <span m="333770">doing</span> <span m="334030">an</span> <span m="334100">induction?</span> <span m="335370">All</span> <span m="335490">right.</span> <span m="335800">So</span> <span m="336210">let's</span> <span m="336430">say</span> <span m="336530">the</span> <span m="336660">boundary</span> <span m="337070">condition,</span> <span m="345280">and</span> <span m="345460">this</span> <span m="345660">always</span> <span m="346060">is</span> <span m="346160">important</span> <span m="346415">with</span> <span m="346670">recurrences,</span> <span m="348170">is</span> <span m="348640">that</span> <span m="348730">the</span> <span m="348820">first</span> <span m="349240">professor</span> <span m="350760">is</span> <span m="350880">hired</span> <span m="351440">in</span> <span m="351570">year</span> <span m="351760">1,</span> <span m="353450">and</span> <span m="353590">there</span> <span m="353680">were</span> <span m="353770">none</span> <span m="354060">before</span> <span m="354360">that</span> <span m="354560">person.</span> <span m="359218">All right?</span> <span m="359690">So</span> <span m="359730">in</span> <span m="359810">year</span> <span m="360020">1</span> <span m="360260">there's</span> <span m="360430">1</span> <span m="360860">professor,</span> <span m="361620">and</span> <span m="361740">that</span> <span m="361870">professor's</span> <span m="362125">new.</span> </p>
<p><span m="364890">OK.</span> <span m="365180">So</span> <span m="365310">now</span> <span m="365550">we</span> <span m="365630">have</span> <span m="365800">all</span> <span m="366020">the</span> <span m="366110">information</span> <span m="366650">necessary</span> <span m="367220">to</span> <span m="367300">solve</span> <span m="367620">the</span> <span m="367700">problem.</span> <span m="368990">So</span> <span m="369440">let's</span> <span m="369690">do</span> <span m="369870">that</span> <span m="370120">and</span> <span m="370330">set</span> <span m="370520">up</span> <span m="370640">a</span> <span m="370710">recurrence.</span> <span m="376290">We're</span> <span m="376440">going</span> <span m="376670">to</span> <span m="377300">define</span> <span m="379450">f</span> <span m="379760">of</span> <span m="379880">n</span> <span m="380990">to</span> <span m="381070">be</span> <span m="381190">the</span> <span m="381320">number</span> <span m="382730">of</span> <span m="382830">professors</span> <span m="385770">during</span> <span m="386110">year</span> <span m="386270">n.</span> <span m="391880">And</span> <span m="392120">we</span> <span m="392260">know</span> <span m="393380">from</span> <span m="393590">our</span> <span m="393710">boundary</span> <span m="394100">condition</span> <span m="394720">that</span> <span m="394850">in</span> <span m="394960">year</span> <span m="395220">0</span> <span m="395700">there</span> <span m="395830">were</span> <span m="395950">none,</span> <span m="397690">and</span> <span m="397870">in</span> <span m="397940">year</span> <span m="398220">1</span> <span m="398790">there</span> <span m="398910">was</span> <span m="399070">1,</span> <span m="400670">a</span> <span m="400770">new</span> <span m="400970">one.</span> <span m="403330">What</span> <span m="403510">is</span> <span m="404140">f</span> <span m="404360">of</span> <span m="404480">2?</span> <span m="407230">How</span> <span m="407410">many</span> <span m="407620">professors</span> <span m="408270">are</span> <span m="408530">there</span> <span m="408790">in</span> <span m="408870">year</span> <span m="409200">2?</span> </p>
<p><span m="410360">AUDIENCE: 1.</span> </p>
<p><span m="411340">PROFESSOR: 1,</span> <span m="411980">because</span> <span m="412500">the</span> <span m="412590">one</span> <span m="412760">that</span> <span m="412840">was</span> <span m="412950">there</span> <span m="413160">was</span> <span m="413280">too</span> <span m="413430">young</span> <span m="413690">to</span> <span m="413770">do</span> <span m="413970">anything.</span> <span m="414350">So</span> <span m="414550">he</span> <span m="415240">or</span> <span m="415440">she</span> <span m="415660">is</span> <span m="415780">the</span> <span m="415880">only</span> <span m="416080">one</span> <span m="416250">left,</span> <span m="417716">all</span> <span m="418140">right,</span> <span m="418400">in</span> <span m="418490">year</span> <span m="418740">2.</span> <span m="419540">What</span> <span m="419860">is</span> <span m="420610">f</span> <span m="420850">of</span> <span m="421050">3?</span> <span m="421370">How</span> <span m="421540">many</span> <span m="421770">profs</span> <span m="422270">are</span> <span m="422500">there</span> <span m="422760">in</span> <span m="422830">year</span> <span m="423050">3?</span> <span m="425000">2--</span> <span m="425640">the</span> <span m="425740">one</span> <span m="426080">you</span> <span m="426170">had,</span> <span m="427060">and</span> <span m="427220">by</span> <span m="427350">that</span> <span m="427610">point</span> <span m="428010">that</span> <span m="428240">one</span> <span m="428380">is</span> <span m="428490">old</span> <span m="428680">enough</span> <span m="428870">to</span> <span m="428990">produce</span> <span m="429340">a</span> <span m="429390">new</span> <span m="429580">one.</span> <span m="432045">All</span> <span m="432490">right.</span> <span m="432590">What's</span> <span m="433080">f</span> <span m="433380">of</span> <span m="433600">4?</span> </p>
<p><span m="436542">AUDIENCE: 3.</span> </p>
<p><span m="438360">PROFESSOR: 3,</span> <span m="439090">the</span> <span m="439220">2</span> <span m="439460">you</span> <span m="439640">had,</span> <span m="440580">and</span> <span m="440790">there</span> <span m="440890">was</span> <span m="441060">one</span> <span m="441510">who's</span> <span m="441720">been</span> <span m="441910">there</span> <span m="442060">2</span> <span m="442240">years</span> <span m="442530">to</span> <span m="442620">produce</span> <span m="442970">a</span> <span m="443030">new</span> <span m="443220">one.</span> <span m="448060">All</span> <span m="448170">right.</span> <span m="448400">And</span> <span m="448550">let's</span> <span m="448750">do</span> <span m="448880">one</span> <span m="449070">more,</span> <span m="449950">f</span> <span m="450190">of</span> <span m="450350">5?</span> </p>
<p><span m="453056">AUDIENCE: 5.</span> </p>
<p><span m="453507">PROFESSOR: 5.</span> <span m="455050">These</span> <span m="455300">guys</span> <span m="455640">produced</span> <span m="456100">1</span> <span m="456340">each,</span> <span m="457070">and</span> <span m="457260">you</span> <span m="457340">had</span> <span m="457680">3</span> <span m="458620">existing.</span> <span m="461120">All</span> <span m="461380">right.</span> <span m="461770">So</span> <span m="462690">we</span> <span m="462860">can</span> <span m="463010">actually</span> <span m="463280">write</span> <span m="463520">down</span> <span m="463730">the</span> <span m="463820">recurrence</span> <span m="464400">now</span> <span m="464680">by</span> <span m="465882">sort</span> <span m="466350">of</span> <span m="466390">the</span> <span m="466480">process</span> <span m="466910">we</span> <span m="467010">just</span> <span m="467300">went</span> <span m="467510">through.</span> <span m="469840">For</span> <span m="470080">years</span> <span m="471540">2</span> <span m="471850">and</span> <span m="471950">beyond,</span> <span m="473550">the</span> <span m="473650">number</span> <span m="473920">of</span> <span m="473980">professors</span> <span m="474530">in</span> <span m="474630">year</span> <span m="474890">n</span> <span m="475850">is</span> <span m="476170">the</span> <span m="476240">number</span> <span m="476530">that</span> <span m="476660">we</span> <span m="476780">had</span> <span m="477110">last</span> <span m="477490">year--</span> <span m="479010">that's</span> <span m="479240">the</span> <span m="479350">previous</span> <span m="479810">ones--</span> <span m="482100">plus</span> <span m="483390">the</span> <span m="483520">number</span> <span m="483870">that</span> <span m="484050">were</span> <span m="484180">generated,</span> <span m="485190">new</span> <span m="485440">professors,</span> <span m="486040">the</span> <span m="486120">graduate</span> <span m="486610">students</span> <span m="487000">that</span> <span m="487250">graduated.</span> <span m="488900">In</span> <span m="489010">terms</span> <span m="489330">of</span> <span m="489440">f,</span> <span m="489840">how</span> <span m="490050">many</span> <span m="490280">new</span> <span m="490510">ones</span> <span m="490780">are</span> <span m="490950">there?</span> </p>
<p><span m="491250">AUDIENCE: f of</span> <span m="491645">n</span> <span m="492040">minus</span> <span m="492435">2.</span> </p>
<p><span m="492830">PROFESSOR: f</span> <span m="493120">of</span> <span m="493240">n</span> <span m="493400">minus</span> <span m="493760">2,</span> <span m="494180">because</span> <span m="494320">that's</span> <span m="494580">the</span> <span m="494670">number</span> <span m="494930">of</span> <span m="494980">professors</span> <span m="495520">that</span> <span m="495590">were</span> <span m="495720">there</span> <span m="496040">a</span> <span m="496080">couple</span> <span m="496330">years</span> <span m="496630">ago,</span> <span m="496860">and</span> <span m="496940">they</span> <span m="497430">are</span> <span m="497520">now</span> <span m="497740">generating</span> <span m="498320">them.</span> <span m="499150">So</span> <span m="499280">that's</span> <span m="499470">the</span> <span m="499580">new</span> <span m="499800">ones</span> <span m="501240">there.</span> <span m="503200">Do</span> <span m="503310">people</span> <span m="503690">recognize</span> <span m="504310">that</span> <span m="504490">recurrence?</span> <span m="506720">Yeah, it's</span> <span m="507090">pretty</span> <span m="507340">famous.</span> <span m="508390">How</span> <span m="508550">many</span> <span m="508750">people</span> <span m="509060">have</span> <span m="509230">not</span> <span m="509540">seen?</span> <span m="509810">This</span> <span m="509960">is</span> <span m="510080">called</span> <span m="510310">the</span> <span m="510390">Fibonacci</span> <span m="511140">recurrence.</span> <span m="511525">It</span> <span m="511910">produces</span> <span m="512320">the</span> <span m="512400">Fibonacci</span> <span m="513000">numbers.</span> <span m="513720">How</span> <span m="513740">many</span> <span m="513919">people</span> <span m="514159">have</span> <span m="514260">not</span> <span m="514690">seen</span> <span m="514940">it?</span> <span m="516480">Yeah,</span> <span m="516740">very</span> <span m="517169">famous.</span> <span m="517640">Yeah,</span> <span m="517870">pretty</span> <span m="518059">much</span> <span m="518299">everybody</span> <span m="518559">has</span> <span m="518820">seen</span> <span m="519179">that.</span> </p>
<p><span m="520780">Actually,</span> <span m="521280">this</span> <span m="521539">is</span> <span m="521650">the</span> <span m="521799">first</span> <span m="522510">recurrence</span> <span m="523280">that</span> <span m="523390">was</span> <span m="523590">known</span> <span m="524090">to</span> <span m="524169">be</span> <span m="524330">studied</span> <span m="525510">of</span> <span m="525650">all</span> <span m="525900">recurrences.</span> <span m="527020">It</span> <span m="527190">was</span> <span m="527410">published</span> <span m="528190">by</span> <span m="528440">Leonardo</span> <span m="529260">Fibonacci</span> <span m="530000">of</span> <span m="530090">Pisa</span> <span m="531260">in</span> <span m="531430">1202,</span> <span m="533425">all right, so</span> <span m="533880">over</span> <span m="534110">800</span> <span m="534490">years</span> <span m="534780">ago,</span> <span m="535620">and</span> <span m="535860">he</span> <span m="536050">studied</span> <span m="536580">it</span> <span m="537180">for</span> <span m="537700">modeling</span> <span m="538270">the</span> <span m="538380">population</span> <span m="539120">growth</span> <span m="539470">of</span> <span m="539570">rabbits.</span> <span m="540680">And</span> <span m="540820">the</span> <span m="540970">idea</span> <span m="541490">is</span> <span m="541910">that</span> <span m="542540">you</span> <span m="542640">have</span> <span m="542760">a</span> <span m="542820">pair</span> <span m="543180">of</span> <span m="543240">rabbits,</span> <span m="544450">and</span> <span m="544870">in</span> <span m="544990">every</span> <span m="545280">month</span> <span m="545630">after</span> <span m="545990">their</span> <span m="546170">first</span> <span m="546650">year</span> <span m="546880">of</span> <span m="546970">life--</span> <span m="547400">sorry,</span> <span m="547630">first</span> <span m="547920">month</span> <span m="548190">of</span> <span m="548310">life--</span> <span m="548910">they</span> <span m="549050">produce</span> <span m="549410">two</span> <span m="549640">new</span> <span m="549900">rabbits.</span> </p>
<p><span m="551330">All</span> <span m="551440">right?</span> <span m="551720">So</span> <span m="551830">it's</span> <span m="551900">the</span> <span m="552000">same</span> <span m="552350">notion</span> <span m="552720">as</span> <span m="552850">here.</span> <span m="553030">The</span> <span m="553150">first</span> <span m="553530">pair</span> <span m="553830">you</span> <span m="553910">do</span> <span m="554030">nothing,</span> <span m="554800">but</span> <span m="554930">after</span> <span m="555190">that</span> <span m="555490">you're</span> <span m="555630">reproducing</span> <span m="556230">one</span> <span m="556440">for</span> <span m="556550">one.</span> <span m="558450">And</span> <span m="558560">that's</span> <span m="559400">an</span> <span m="559510">abstraction,</span> <span m="560150">but</span> <span m="560330">it</span> <span m="560480">produces</span> <span m="561050">the</span> <span m="561170">same</span> <span m="561820">recurrence.</span> </p>
<p><span m="564170">Now,</span> <span m="564480">Fibonacci</span> <span m="565170">is</span> <span m="565330">credited</span> <span m="566060">with</span> <span m="566190">discovering</span> <span m="566840">it.</span> <span m="567440">Really</span> <span m="567830">that</span> <span m="568050">means</span> <span m="568330">he's</span> <span m="568570">the</span> <span m="568680">one</span> <span m="568930">that</span> <span m="569040">told</span> <span m="569310">the</span> <span m="569400">Europeans</span> <span m="570010">about</span> <span m="570370">it</span> <span m="570610">back</span> <span m="570890">then.</span> <span m="571680">And</span> <span m="571830">in</span> <span m="571900">fact,</span> <span m="572270">it's</span> <span m="572410">now</span> <span m="572610">been</span> <span m="572780">traced</span> <span m="573330">back</span> <span m="573680">in</span> <span m="573880">to</span> <span m="574590">about</span> <span m="574880">200</span> <span m="575260">BC.</span> <span m="575800">The</span> <span m="575990">Indian</span> <span m="576390">mathematicians</span> <span m="577550">knew</span> <span m="577670">all</span> <span m="577900">about</span> <span m="578250">Fibonacci's</span> <span m="579090">recurrence,</span> <span m="580730">and</span> <span m="580830">they</span> <span m="580950">were</span> <span m="581060">using</span> <span m="581370">it</span> <span m="581470">to</span> <span m="581560">study</span> <span m="582330">certain</span> <span m="582630">properties</span> <span m="583120">of</span> <span m="583230">grammar</span> <span m="583770">and</span> <span m="583900">music</span> <span m="584410">way</span> <span m="584740">back</span> <span m="585200">at</span> <span m="585330">200</span> <span m="585660">BC.</span> </p>
<p><span m="587190">This</span> <span m="587470">recurrence</span> <span m="587950">comes</span> <span m="588150">up</span> <span m="588270">in</span> <span m="588360">all</span> <span m="588560">sorts</span> <span m="588830">of</span> <span m="588940">applications.</span> <span m="590150">Kepler</span> <span m="590620">used</span> <span m="590950">it</span> <span m="591040">in</span> <span m="591120">the</span> <span m="591210">16th</span> <span m="591750">century</span> <span m="592710">while</span> <span m="593460">studying</span> <span m="593890">how</span> <span m="594630">the</span> <span m="594760">leaves</span> <span m="595170">of</span> <span m="595240">a</span> <span m="595330">flower</span> <span m="595900">are</span> <span m="596170">arranged</span> <span m="596540">around</span> <span m="596940">the</span> <span m="597110">stem--</span> <span m="597810">how</span> <span m="597960">many</span> <span m="598170">leaves</span> <span m="598460">you</span> <span m="598550">have  in</span> <span m="598900">sort</span> <span m="599050">of</span> <span m="599130">each</span> <span m="599320">level</span> <span m="599580">coming</span> <span m="599870">out</span> <span m="600020">around a</span> <span m="600380">stem.</span> <span m="602270">The</span> <span m="603300">first</span> <span m="603670">solution</span> <span m="604210">was</span> <span m="604350">discovered</span> <span m="604840">by</span> <span m="605050">de</span> <span m="605280">Moivre</span> <span m="605820">in</span> <span m="605890">the</span> <span m="606050">18th</span> <span m="606450">century.</span> <span m="606980">And we're</span> <span m="607120">going</span> <span m="607200">to</span> <span m="607250">talk</span> <span m="607500">about</span> <span m="607670">how</span> <span m="607790">to</span> <span m="607870">solve</span> <span m="608200">this</span> <span m="608360">in</span> <span m="608460">a</span> <span m="608500">minute.</span> <span m="608930">But</span> <span m="609030">he</span> <span m="609040">was</span> <span m="609190">the</span> <span m="609280">first</span> <span m="609560">one</span> <span m="609680">to</span> <span m="609750">figure</span> <span m="610020">out</span> <span m="610100">a</span> <span m="610180">closed</span> <span m="610730">form</span> <span m="611500">expression</span> <span m="612220">for</span> <span m="612390">f</span> <span m="612520">of</span> <span m="612640">n.</span> </p>
<p><span m="613840">Lame</span> <span m="614230">used it</span> <span m="614530">in</span> <span m="614650">the</span> <span m="614730">19th</span> <span m="615200">century</span> <span m="616090">when</span> <span m="616270">he</span> <span m="616330">was</span> <span m="616490">studying</span> <span m="616950">the</span> <span m="617050">Euclidean</span> <span m="617600">GCD</span> <span m="618240">algorithm.</span> <span m="619170">You</span> <span m="619240">know</span> <span m="619330">that</span> <span m="619530">pulverizer</span> <span m="620230">thing</span> <span m="620530">and</span> <span m="620620">doing</span> <span m="620800">GCDs?</span> <span m="621990">It</span> <span m="622130">turns</span> <span m="622450">out</span> <span m="622840">that</span> <span m="623440">if</span> <span m="623560">you</span> <span m="623640">want</span> <span m="623770">to</span> <span m="623820">analyze</span> <span m="624270">the</span> <span m="624380">running</span> <span m="624690">time,</span> <span m="625690">well,</span> <span m="626120">you</span> <span m="626260">get</span> <span m="626400">a</span> <span m="626450">Fibonacci</span> <span m="627020">recurrence</span> <span m="627640">comes</span> <span m="627920">into</span> <span m="628130">play</span> <span m="628410">there,</span> <span m="629110">and</span> <span m="629240">that</span> <span m="629400">was</span> <span m="629530">discovered</span> <span m="629960">in</span> <span m="630010">the</span> <span m="630090">19th</span> <span m="630470">century.</span> </p>
<p><span m="631720">In</span> <span m="631820">the</span> <span m="631900">20th</span> <span m="632300">century, it</span> <span m="632800">was</span> <span m="632960">used</span> <span m="633390">in</span> <span m="633450">the</span> <span m="633540">study</span> <span m="633910">of</span> <span m="634030">optics,</span> <span m="634980">economics,</span> <span m="636200">and</span> <span m="636480">algorithms,</span> <span m="637220">and</span> <span m="637350">it</span> <span m="637390">was</span> <span m="637590">named</span> <span m="638030">for</span> <span m="638180">Fibonacci.</span> <span m="638470">It</span> <span m="638760">got</span> <span m="638920">a</span> <span m="638970">name</span> <span m="639390">in</span> <span m="639460">the</span> <span m="639520">19th</span> <span m="639920">century.</span> <span m="641540">In</span> <span m="641570">fact,</span> <span m="641970">this</span> <span m="642130">is</span> <span m="642220">so</span> <span m="642600">popular</span> <span m="643240">and</span> <span m="643310">used</span> <span m="643570">in</span> <span m="643660">so</span> <span m="643850">many</span> <span m="644020">places</span> <span m="644990">there</span> <span m="645180">is</span> <span m="645270">a</span> <span m="645370">journal</span> <span m="646030">in</span> <span m="646160">mathematics</span> <span m="646880">called</span> <span m="647200">the</span> <span m="647720"><i>Fibonacci</i></span> <span m="648410"><i>Quarterly</i></span> <span m="648920"><i>Journal</i></span> <span m="649320">where</span> <span m="649420">they</span> <span m="649560">study</span> <span m="650790">these</span> <span m="651220">kinds</span> <span m="651440">of</span> <span m="651550">things.</span> </p>
<p><span m="653190">So</span> <span m="653320">today</span> <span m="653720">what</span> <span m="653840">we're</span> <span m="653930">going</span> <span m="654020">to</span> <span m="654060">do</span> <span m="654180">is</span> <span m="654280">solve</span> <span m="654630">this</span> <span m="654790">recurrence</span> <span m="655420">and</span> <span m="655540">actually</span> <span m="655830">solve</span> <span m="656150">a</span> <span m="656210">much</span> <span m="656470">broader</span> <span m="656920">family</span> <span m="657430">of</span> <span m="657500">recurrences</span> <span m="658060">called</span> <span m="658780">linear</span> <span m="659150">recurrences.</span> <span m="661370">And</span> <span m="661510">we're</span> <span m="661610">going</span> <span m="661720">to</span> <span m="661760">get</span> <span m="661920">a</span> <span m="661980">closed</span> <span m="662370">form.</span> <span m="662980">I</span> <span m="663030">mean,</span> <span m="663150">you</span> <span m="663270">can</span> <span m="663400">produce</span> <span m="663760">the</span> <span m="663810">Fibonacci</span> <span m="664350">numbers</span> <span m="664600">one</span> <span m="664840">after</span> <span m="665110">another,</span> <span m="665960">but</span> <span m="666090">we're</span> <span m="666250">going</span> <span m="666350">to</span> <span m="666430">derive</span> <span m="666840">a</span> <span m="666910">formula</span> <span m="667880">for</span> <span m="668000">the</span> <span m="668160">n-th</span> <span m="668360">Fibonacci</span> <span m="670040">number.</span> </p>
<p><span m="671170">And when</span> <span m="671600">we're</span> <span m="671800">going</span> <span m="671880">to</span> <span m="671960">do</span> <span m="672230">it,</span> <span m="672400">we're</span> <span m="672540">going</span> <span m="672640">to</span> <span m="672760">do</span> <span m="673000">it</span> <span m="673330">more</span> <span m="673590">broadly</span> <span m="674730">for</span> <span m="675020">a</span> <span m="675060">class</span> <span m="675490">of</span> <span m="675590">linear</span> <span m="676030">recurrences.</span> <span m="676710">So</span> <span m="676880">let</span> <span m="676980">me</span> <span m="677070">define</span> <span m="677365">what</span> <span m="677660">that</span> <span m="677900">is.</span> <span m="683950">So</span> <span m="684210">a</span> <span m="684290">recurrence</span> <span m="686070">is</span> <span m="686360">said</span> <span m="686490">to</span> <span m="686580">be</span> <span m="686720">linear</span> <span m="692740">if</span> <span m="692960">it</span> <span m="693040">is</span> <span m="693170">of</span> <span m="693290">the</span> <span m="693380">form</span> <span m="702340">f</span> <span m="702570">of</span> <span m="702690">n</span> <span m="704150">equals</span> <span m="704680">a</span> <span m="704740">constant</span> <span m="705520">a1</span> <span m="706820">times</span> <span m="707500">f</span> <span m="707970">of n</span> <span m="708440">minus</span> <span m="708910">1</span> <span m="710150">plus</span> <span m="710670">a2</span> <span m="712320">times</span> <span m="712780">f</span> <span m="713010">of</span> <span m="713130">n</span> <span m="713310">minus</span> <span m="713750">2</span> <span m="714830">plus</span> <span m="715160">dot</span> <span m="715370">dot</span> <span m="715570">dot</span> <span m="717090">d'th</span> <span m="717970">constant</span> <span m="718470">ad</span> <span m="719396">times</span> <span m="719860">f</span> <span m="720030">of</span> <span m="720170">n</span> <span m="720330">minus</span> <span m="720810">d.</span> </p>
<p><span m="722800">And</span> <span m="723130">we</span> <span m="723230">could</span> <span m="723370">simplify</span> <span m="723900">that</span> <span m="724150">as</span> <span m="724450">the</span> <span m="724550">sum</span> <span m="726440">i</span> <span m="726710">equals</span> <span m="727110">1</span> <span m="727410">to</span> <span m="727520">d</span> <span m="729011">of</span> <span m="729850">a</span> <span m="730060">sub</span> <span m="730300">i</span> <span m="730950">times</span> <span m="731460">f</span> <span m="731780">of</span> <span m="732100">n</span> <span m="732380">minus</span> <span m="732770">i.</span> <span m="734590">And</span> <span m="735070">the</span> <span m="735150">constants</span> <span m="736520">are</span> <span m="736690">fixed</span> <span m="737787">here,</span> <span m="739190">so</span> <span m="739380">for</span> <span m="739660">fixed</span> <span m="742030">a</span> <span m="742500">sub</span> <span m="742730">i</span> <span m="743500">and</span> <span m="743760">d.</span> <span m="745280">So</span> <span m="745420">the</span> <span m="745530">number</span> <span m="745890">of</span> <span m="745960">terms</span> <span m="746450">has</span> <span m="746630">to</span> <span m="746710">be</span> <span m="746760">a</span> <span m="746810">constant,</span> <span m="747420">and</span> <span m="747450">each</span> <span m="748380">coefficient</span> <span m="749000">has</span> <span m="749240">to</span> <span m="749310">be</span> <span m="749400">a</span> <span m="749470">constant.</span> <span m="750510">Can't</span> <span m="750780">vary.</span> </p>
<p><span m="752030">And</span> <span m="752360">we</span> <span m="752460">define</span> <span m="753160">d</span> <span m="753580">to</span> <span m="753660">be</span> <span m="753790">the</span> <span m="754000">order</span> <span m="754440">of</span> <span m="754570">the</span> <span m="754660">recurrence.</span> <span m="756870">So</span> <span m="756940">d</span> <span m="757830">is</span> <span m="757990">the</span> <span m="758190">order</span> <span m="759460">of</span> <span m="759610">the</span> <span m="759810">recurrence.</span> <span m="762820">OK?</span> <span m="764790">And</span> <span m="765000">you</span> <span m="765090">can</span> <span m="765210">see,</span> <span m="765400">of</span> <span m="765470">course,</span> <span m="765850">that</span> <span m="766180">Fibonacci's</span> <span m="766740">recurrence</span> <span m="767095">is</span> <span m="767350">linear.</span> <span m="768660">What's</span> <span m="768870">its</span> <span m="769020">order?</span> <span m="771080">2.</span> <span m="772190">And</span> <span m="772610">the</span> <span m="772710">coefficients,</span> <span m="773390">the</span> <span m="773550">a's,</span> <span m="773880">are</span> <span m="774220">just</span> <span m="774460">1.</span> <span m="774880">So</span> <span m="775030">it's</span> <span m="775140">a</span> <span m="775190">simple</span> <span m="775550">linear</span> <span m="775810">recurrence.</span> </p>
<p><span m="778620">All</span> <span m="778730">right.</span> <span m="778910">So</span> <span m="779000">let's</span> <span m="779220">see</span> <span m="780250">how</span> <span m="780390">to</span> <span m="780490">solve it. Well,</span> <span m="780940">actually,</span> <span m="781160">before</span> <span m="781450">we do</span> <span m="781550">that,</span> <span m="782330">can</span> <span m="782450">you</span> <span m="782530">see</span> <span m="782690">the</span> <span m="782820">difference</span> <span m="783240">between</span> <span m="784460">this</span> <span m="784860">recurrence,</span> <span m="785390">linear,</span> <span m="786120">and</span> <span m="786330">divide-and-conquer</span> <span m="787660">recurrences?</span> <span m="791180">Right?</span> <span m="793050">What</span> <span m="793190">do</span> <span m="793280">I</span> <span m="793390">have</span> <span m="793730">here</span> <span m="794230">inside</span> <span m="794800">for a</span> <span m="794930">divide-and-conquer</span> <span m="795720">recurrence?</span> <span m="799400">I</span> <span m="799470">get</span> <span m="799600">a</span> <span m="799650">fraction</span> <span m="800190">of</span> <span m="800280">n,</span> <span m="800500">right?</span> </p>
<p><span m="801640">And</span> <span m="801760">here,</span> <span m="802180">I'm</span> <span m="802270">subtracting</span> <span m="803060">a</span> <span m="803120">constant</span> <span m="803630">from</span> <span m="803710">n,</span> <span m="804190">usually like</span> <span m="804510">1,</span> <span m="804750">2,</span> <span m="804930">3,</span> <span m="805180">an integer</span> <span m="805550">from n.</span> <span m="806730">So</span> <span m="806930">linear</span> <span m="807430">is</span> <span m="807530">when</span> <span m="807690">inside</span> <span m="808150">you</span> <span m="808230">have</span> <span m="808450">n</span> <span m="808670">minus</span> <span m="809300">1,</span> <span m="809660">n</span> <span m="809780">minus</span> <span m="810170">2.</span> <span m="810940">Divide and</span> <span m="811360">conquer, you</span> <span m="811820">got</span> <span m="811950">n/2</span> <span m="812810">or</span> <span m="812920">3/4</span> <span m="813225">n.</span> <span m="813945">And</span> <span m="814360">it</span> <span m="814480">makes</span> <span m="814690">a</span> <span m="814740">huge</span> <span m="815110">difference</span> <span m="816360">in</span> <span m="816500">the</span> <span m="816600">solution.</span> </p>
<p><span m="817992">All</span> <span m="818430">right.</span> <span m="818780">So</span> <span m="819490">what</span> <span m="819760">I'm</span> <span m="819840">going</span> <span m="819930">to</span> <span m="819990">do</span> <span m="820170">is</span> <span m="820280">give</span> <span m="820420">you a</span> <span m="820570">closed-form</span> <span m="821390">solution</span> <span m="821970">for</span> <span m="822060">these</span> <span m="822240">kinds</span> <span m="822470">of</span> <span m="822560">recurrences.</span> <span m="822820">And</span> <span m="823080">it</span> <span m="823240">won't</span> <span m="823520">be</span> <span m="823920">completely</span> <span m="824460">easy</span> <span m="825790">because</span> <span m="826400">it</span> <span m="826550">took</span> <span m="826980">Europeans</span> <span m="827690">six</span> <span m="828095">centuries</span> <span m="828880">to</span> <span m="828980">find</span> <span m="829420">the</span> <span m="829510">solution</span> <span m="830350">to</span> <span m="830500">this.</span> <span m="831660">Right?</span> <span m="832000">Fibonacci</span> <span m="832490">discovers</span> <span m="833020">the</span> <span m="833090">thing</span> <span m="833340">in</span> <span m="833420">1200</span> <span m="833720">and</span> <span m="834150">tells</span> <span m="834390">everybody</span> <span m="834700">about</span> <span m="834990">it.</span> <span m="835210">And</span> <span m="835540">it</span> <span m="835740">wasn't</span> <span m="836060">until</span> <span m="837230">600</span> <span m="837670">years</span> <span m="837900">later</span> <span m="838305">that</span> <span m="838710">they</span> <span m="838930">figured</span> <span m="839250">out</span> <span m="839670">a</span> <span m="839770">closed-form</span> <span m="840320">solution.</span> </p>
<p><span m="843824">So</span> <span m="844310">let's</span> <span m="844540">do</span> <span m="844650">that.</span> <span m="848040">Now,</span> <span m="848880">what</span> <span m="849180">we're</span> <span m="849300">going</span> <span m="849420">to</span> <span m="849510">do</span> <span m="850450">to</span> <span m="850590">do</span> <span m="850770">it</span> <span m="850850">this</span> <span m="851020">first</span> <span m="851270">time</span> <span m="851480">ourselves,</span> <span m="851770">and</span> <span m="852060">we</span> <span m="852240">don't</span> <span m="852440">have</span> <span m="852610">the</span> <span m="852680">formula,</span> <span m="853610">is</span> <span m="853780">to</span> <span m="853860">use</span> <span m="854140">guess</span> <span m="854410">and</span> <span m="854520">verify.</span> <span m="855750">So</span> <span m="855870">we're</span> <span m="856000">going</span> <span m="856100">to</span> <span m="856150">guess</span> <span m="856580">a</span> <span m="856640">solution</span> <span m="857620">and</span> <span m="857760">check</span> <span m="858020">that</span> <span m="858120">it</span> <span m="858200">works.</span> <span m="859750">And</span> <span m="859950">we're</span> <span m="860090">going</span> <span m="860210">to</span> <span m="860480">guess,</span> <span m="860790">really,</span> <span m="861050">a</span> <span m="861120">class</span> <span m="861600">of</span> <span m="861680">solutions.</span> <span m="863030">We're</span> <span m="863180">going</span> <span m="863260">to</span> <span m="863360">try</span> <span m="865560">f</span> <span m="865900">of</span> <span m="866020">n</span> <span m="867320">is</span> <span m="868530">an</span> <span m="868670">exponential</span> <span m="869320">in</span> <span m="869420">n,</span> <span m="869860">alpha to the</span> <span m="870180">n</span> <span m="870770">for</span> <span m="870890">some</span> <span m="871280">constant</span> <span m="871800">alpha.</span> </p>
<p><span m="878310">Now,</span> <span m="878940">we're</span> <span m="879140">going</span> <span m="879240">to</span> <span m="879700">figure</span> <span m="880210">out</span> <span m="880430">what</span> <span m="880590">alpha</span> <span m="880900">is</span> <span m="881220">as</span> <span m="881340">we</span> <span m="881470">go</span> <span m="881620">along</span> <span m="882340">during</span> <span m="882560">the</span> <span m="882640">verification</span> <span m="883430">process.</span> <span m="885330">All</span> <span m="885440">right?</span> <span m="885630">So</span> <span m="885730">let's</span> <span m="886240">try</span> <span m="886470">to</span> <span m="886550">verify</span> <span m="887130">this</span> <span m="887380">guess</span> <span m="887790">and</span> <span m="887910">plug</span> <span m="888180">it</span> <span m="888240">in.</span> <span m="890830">We</span> <span m="890940">know</span> <span m="891200">that</span> <span m="891430">f</span> <span m="891660">of</span> <span m="891790">n</span> <span m="892050">is</span> <span m="892340">f</span> <span m="892550">of</span> <span m="892650">n</span> <span m="892800">minus</span> <span m="893220">1</span> <span m="893620">plus</span> <span m="894660">f</span> <span m="894860">of</span> <span m="894970">n</span> <span m="895130">minus</span> <span m="895490">2.</span> <span m="896595">Well,</span> <span m="896860">let's</span> <span m="897290">plug</span> <span m="897680">that</span> <span m="897870">in</span> <span m="898020">and</span> <span m="898150">see</span> <span m="898320">what</span> <span m="898460">we</span> <span m="898560">get.</span> <span m="900120">That</span> <span m="900290">gives</span> <span m="900480">us</span> <span m="900630">alpha</span> <span m="900940">to</span> <span m="900990">the</span> <span m="901130">n</span> <span m="901630">equals</span> <span m="902630">alpha to the</span> <span m="903000">n</span> <span m="903310">minus</span> <span m="903670">1</span> <span m="904210">plus</span> <span m="905070">alpha</span> <span m="905300">to</span> <span m="905500">the</span> <span m="905660">n</span> <span m="905770">minus</span> <span m="906120">2.</span> <span m="908880">All</span> <span m="909320">right?</span> </p>
<p><span m="909710">Now,</span> <span m="911640">that</span> <span m="911930">means</span> <span m="913190">I</span> <span m="913280">can</span> <span m="913440">divide</span> <span m="914020">by</span> <span m="914180">alpha to</span> <span m="914450">the</span> <span m="914580">n</span> <span m="914690">minus</span> <span m="915050">2,</span> <span m="915300">and</span> <span m="915420">I</span> <span m="915490">get</span> <span m="916360">alpha</span> <span m="916690">squared</span> <span m="917300">equals</span> <span m="917820">alpha</span> <span m="918260">plus</span> <span m="918590">1.</span> <span m="920920">And</span> <span m="921080">now</span> <span m="921390">I</span> <span m="921470">can</span> <span m="921680">use</span> <span m="922140">the</span> <span m="922220">quadratic</span> <span m="922720">formula</span> <span m="923200">to</span> <span m="923310">solve</span> <span m="923690">for</span> <span m="923850">alpha.</span> <span m="925312">All right?</span> <span m="925720">That</span> <span m="925920">means</span> <span m="926200">that</span> <span m="927140">alpha</span> <span m="927430">squared</span> <span m="927860">minus</span> <span m="928270">alpha</span> <span m="929850">minus</span> <span m="930230">1</span> <span m="930550">equals</span> <span m="930950">0,</span> <span m="932410">which</span> <span m="932600">means</span> <span m="932910">that</span> <span m="934130">alpha</span> <span m="934620">equals--</span> <span m="937930">minus</span> <span m="938280">minus</span> <span m="938600">1</span> <span m="938760">is</span> <span m="938860">1</span> <span m="939450">plus</span> <span m="939720">or</span> <span m="939790">minus</span> <span m="940260">the</span> <span m="940310">square</span> <span m="940680">root</span> <span m="940870">of--</span> <span m="941820">minus</span> <span m="942110">1</span> <span m="942250">squared</span> <span m="942680">is</span> <span m="942810">1</span> <span m="943730">minus</span> <span m="944110">4ac</span> <span m="945420">plus</span> <span m="945680">4</span> <span m="947290">over</span> <span m="947840">2.</span> <span m="950010">OK?</span> </p>
<p><span m="950460">So</span> <span m="951820">there's</span> <span m="952030">two</span> <span m="952260">possible</span> <span m="952830">solutions</span> <span m="953420">here.</span> <span m="954990">This</span> <span m="955200">is</span> <span m="955340">1</span> <span m="955700">plus</span> <span m="955990">or</span> <span m="956050">minus</span> <span m="956410">the</span> <span m="956480">square</span> <span m="956680">root</span> <span m="956810">of</span> <span m="956900">5</span> <span m="957350">over</span> <span m="957560">2.</span> <span m="960670">So</span> <span m="961010">it</span> <span m="961210">works</span> <span m="961790">if</span> <span m="962010">f</span> <span m="962210">of</span> <span m="962330">n</span> <span m="962550">equals</span> <span m="963980">either--</span> <span m="965100">well,</span> <span m="965470">we'll</span> <span m="965670">call</span> <span m="966010">the</span> <span m="966110">roots</span> <span m="966540">here</span> <span m="967180">alpha</span> <span m="967490">1</span> <span m="967780">and alpha</span> <span m="968070">2.</span> <span m="974870">Alpha</span> <span m="975250">1</span> <span m="975500">will</span> <span m="975590">be</span> <span m="975670">the</span> <span m="975780">positive</span> <span m="976280">case,</span> <span m="978130">1</span> <span m="978370">plus</span> <span m="978680">square</span> <span m="978910">root</span> <span m="978990">of</span> <span m="979080">5</span> <span m="979380">over</span> <span m="979560">2.</span> <span m="981176">And</span> <span m="981590">alpha</span> <span m="981960">2</span> <span m="982260">will</span> <span m="982370">be</span> <span m="982470">the</span> <span m="982570">negative</span> <span m="983010">case,</span> <span m="983640">1</span> <span m="983830">minus</span> <span m="984250">square</span> <span m="984460">root</span> <span m="984570">of</span> <span m="984650">5</span> <span m="985780">over</span> <span m="985990">2.</span> <span m="988600">All right?</span> <span m="989080">So</span> <span m="989430">guess</span> <span m="989760">and</span> <span m="989860">verify</span> <span m="990570">works</span> <span m="991850">so</span> <span m="992040">far</span> <span m="993730">if</span> <span m="994000">we</span> <span m="994150">have</span> <span m="994450">an</span> <span m="994540">exponential</span> <span m="995700">with</span> <span m="996280">either</span> <span m="996530">one</span> <span m="996710">of</span> <span m="996780">those</span> <span m="997000">bases.</span> <span m="999260">All right?</span> </p>
<p><span m="1001570">By</span> <span m="1001730">the</span> <span m="1001830">way,</span> <span m="1001960">does</span> <span m="1002100">anybody</span> <span m="1002470">recognize</span> <span m="1004959">that</span> <span m="1005432">number?</span> <span m="1009220">It's a</span> <span m="1009280">famous</span> <span m="1009730">number.</span> </p>
<p><span m="1010933">AUDIENCE: Golden</span> <span m="1011316">ratio.</span> </p>
<p><span m="1011700">PROFESSOR: The</span> <span m="1011820">golden</span> <span m="1012230">ratio,</span> <span m="1012940">which</span> <span m="1013360">is</span> <span m="1013460">supposed</span> <span m="1013820">to</span> <span m="1013890">have</span> <span m="1014000">all</span> <span m="1014200">these</span> <span m="1014380">magical</span> <span m="1015030">mystical</span> <span m="1015530">properties.</span> <span m="1016230">That</span> <span m="1016870">when</span> <span m="1017010">you</span> <span m="1017170">look</span> <span m="1017400">at</span> <span m="1017460">a</span> <span m="1017510">building,</span> <span m="1018000">if</span> <span m="1018070">its</span> <span m="1018220">aspect</span> <span m="1018800">ratio</span> <span m="1019250">is</span> <span m="1019360">that,</span> <span m="1019630">it's</span> <span m="1020180">perfect</span> <span m="1020650">to</span> <span m="1020740">the</span> <span m="1021090">human</span> <span m="1021310">eye.</span> <span m="1022060">I</span> <span m="1022200">don't</span> <span m="1022340">know.</span> <span m="1023030">But</span> <span m="1023240">there's</span> <span m="1023400">a</span> <span m="1023460">lot</span> <span m="1023660">of</span> <span m="1023720">stuff</span> <span m="1023930">about</span> <span m="1024119">the</span> <span m="1024210">golden</span> <span m="1024480">ratio</span> <span m="1024940">that</span> <span m="1025180">happens</span> <span m="1025589">to</span> <span m="1025670">come</span> <span m="1025849">up</span> <span m="1025990">here.</span> <span m="1027470">OK?</span> </p>
<p><span m="1030240">Now</span> <span m="1030410">in</span> <span m="1030520">fact,</span> <span m="1031530">there's</span> <span m="1031740">more</span> <span m="1032099">than</span> <span m="1032260">just</span> <span m="1032609">these</span> <span m="1032960">two</span> <span m="1033190">solutions.</span> <span m="1034790">It</span> <span m="1034950">turns</span> <span m="1035329">out</span> <span m="1036339">that</span> <span m="1036500">whenever</span> <span m="1036780">you</span> <span m="1036849">have</span> <span m="1037030">a</span> <span m="1037099">linear</span> <span m="1037410">recurrence</span> <span m="1038819">and</span> <span m="1038970">you've</span> <span m="1039079">got</span> <span m="1039380">two</span> <span m="1039560">or</span> <span m="1039619">more</span> <span m="1039859">solutions</span> <span m="1040440">like</span> <span m="1040650">that,</span> <span m="1041810">any</span> <span m="1042200">linear</span> <span m="1042660">combination</span> <span m="1043500">is</span> <span m="1043670">also</span> <span m="1044079">a</span> <span m="1044150">solution.</span> <span m="1045615">All right?</span> <span m="1045930">So we're</span> <span m="1046130">just</span> <span m="1046460">going</span> <span m="1046560">to</span> <span m="1046630">state</span> <span m="1046970">that</span> <span m="1047130">as</span> <span m="1047240">a</span> <span m="1047310">fact.</span> <span m="1049160">It's</span> <span m="1049210">not</span> <span m="1049410">too</span> <span m="1049540">hard</span> <span m="1049760">to</span> <span m="1049820">prove,</span> <span m="1050085">but</span> <span m="1050350">we</span> <span m="1050480">won't</span> <span m="1050820">prove</span> <span m="1051010">it</span> <span m="1051070">in</span> <span m="1051150">class.</span> </p>
<p><span m="1055110">So</span> <span m="1055530">if</span> <span m="1057050">f</span> <span m="1057280">of</span> <span m="1057410">n</span> <span m="1057670">equals</span> <span m="1058090">alpha</span> <span m="1058490">1</span> <span m="1058760">to</span> <span m="1058830">the</span> <span m="1059050">n--</span> <span m="1059420">and</span> <span m="1059750">this</span> <span m="1059920">is</span> <span m="1060000">true</span> <span m="1060180">for</span> <span m="1060340">any</span> <span m="1060590">alpha</span> <span m="1060920">1 and</span> <span m="1061230">alpha</span> <span m="1061390">2--</span> <span m="1063090">and</span> <span m="1063510">f</span> <span m="1063730">of</span> <span m="1063860">n</span> <span m="1064590">equals</span> <span m="1065020">alpha</span> <span m="1065410">2 to the</span> <span m="1065815">n</span> <span m="1067110">are</span> <span m="1067290">solutions</span> <span m="1073200">to</span> <span m="1073360">a</span> <span m="1073440">linear</span> <span m="1073860">recurrence--</span> <span m="1082320">and</span> <span m="1083230">here</span> <span m="1083630">I</span> <span m="1083690">mean</span> <span m="1084030">without</span> <span m="1084460">yet</span> <span m="1084970">applying</span> <span m="1085320">the</span> <span m="1085400">boundary</span> <span m="1085860">conditions.</span> <span m="1086680">So</span> <span m="1087000">far,</span> <span m="1087240">we've</span> <span m="1087410">ignored</span> <span m="1087870">the</span> <span m="1087950">boundary</span> <span m="1088350">conditions,</span> <span m="1088860">and</span> <span m="1089050">we'll</span> <span m="1089210">just</span> <span m="1089690">do</span> <span m="1089800">that</span> <span m="1089980">for</span> <span m="1090050">a</span> <span m="1090080">little</span> <span m="1090270">longer.</span> </p>
<p><span m="1091580">So</span> <span m="1091770">if</span> <span m="1091840">there</span> <span m="1091960">are</span> <span m="1092020">solutions</span> <span m="1092620">without</span> <span m="1093300">worrying</span> <span m="1093660">about</span> <span m="1093900">the</span> <span m="1093980">boundary</span> <span m="1094480">conditions,</span> <span m="1101230">then</span> <span m="1103530">f</span> <span m="1103760">of</span> <span m="1103880">n</span> <span m="1104710">equals</span> <span m="1105150">c1</span> <span m="1106170">times</span> <span m="1106630">alpha</span> <span m="1107020">1</span> <span m="1107210">to</span> <span m="1107300">the</span> <span m="1107470">n</span> <span m="1107920">plus</span> <span m="1108570">c2</span> <span m="1109730">alpha 2 to the</span> <span m="1110130">n</span> <span m="1111260">is</span> <span m="1111460">also</span> <span m="1111850">a</span> <span m="1111900">solution</span> <span m="1112420">for</span> <span m="1112560">any</span> <span m="1112820">constants</span> <span m="1113390">c1</span> <span m="1114010">and</span> <span m="1114130">c2.</span> <span m="1136380">All</span> <span m="1136450">right?</span> <span m="1136630">So</span> <span m="1136790">any</span> <span m="1137140">linear</span> <span m="1137550">combination</span> <span m="1139080">of</span> <span m="1139230">our</span> <span m="1139370">solutions</span> <span m="1140120">also</span> <span m="1140520">works.</span> <span m="1141320">If</span> <span m="1141460">you</span> <span m="1141580">plugged</span> <span m="1141920">it</span> <span m="1141990">in to</span> <span m="1142210">do</span> <span m="1142340">verify,</span> <span m="1143520">it</span> <span m="1143670">would</span> <span m="1143830">be</span> <span m="1143980">fine.</span> </p>
<p><span m="1145580">So</span> <span m="1146000">that</span> <span m="1146280">means</span> <span m="1146740">that</span> <span m="1150410">f</span> <span m="1150640">of</span> <span m="1150770">n</span> <span m="1151560">equals</span> <span m="1152010">c1</span> <span m="1153520">times</span> <span m="1154380">1</span> <span m="1154630">plus</span> <span m="1154950">square</span> <span m="1155220">root</span> <span m="1155330">of</span> <span m="1155410">5</span> <span m="1156200">over</span> <span m="1156520">2</span> <span m="1156860">to</span> <span m="1156940">the</span> <span m="1157110">n</span> <span m="1158110">plus</span> <span m="1158800">c2</span> <span m="1160130">1</span> <span m="1160410">minus</span> <span m="1160890">square</span> <span m="1161150">root</span> <span m="1161240">of</span> <span m="1161350">5</span> <span m="1162420">over</span> <span m="1162700">2</span> <span m="1163010">to</span> <span m="1163060">the</span> <span m="1163230">n</span> <span m="1165090">is</span> <span m="1165280">a</span> <span m="1165360">solution--</span> <span m="1168650">oops--</span> <span m="1170560">to</span> <span m="1170920">Fibonacci's</span> <span m="1171470">recurrence</span> <span m="1173500">without</span> <span m="1173870">boundary</span> <span m="1174210">conditions</span> <span m="1175000">again.</span> <span m="1181550">All</span> <span m="1181680">right?</span> <span m="1181910">So</span> <span m="1182050">I</span> <span m="1182150">could</span> <span m="1182380">plug</span> <span m="1182810">this</span> <span m="1183220">expression</span> <span m="1184730">into</span> <span m="1185060">the</span> <span m="1185170">recurrence,</span> <span m="1186750">and</span> <span m="1186990">it</span> <span m="1187060">would</span> <span m="1187200">satisfy</span> <span m="1187800">it.</span> <span m="1189885">I</span> <span m="1190150">won't</span> <span m="1190410">do</span> <span m="1190550">that.</span> </p>
<p><span m="1194180">But</span> <span m="1194470">we</span> <span m="1194590">haven't</span> <span m="1195110">dealt</span> <span m="1195310">with</span> <span m="1195400">the</span> <span m="1195480">boundary</span> <span m="1195870">conditions</span> <span m="1196340">yet.</span> <span m="1197790">And</span> <span m="1197890">in</span> <span m="1197970">fact,</span> <span m="1198290">dealing</span> <span m="1198570">with</span> <span m="1198700">the</span> <span m="1198750">boundary</span> <span m="1199200">conditions</span> <span m="1199970">is</span> <span m="1200100">what</span> <span m="1200280">determines</span> <span m="1201640">the</span> <span m="1201740">values</span> <span m="1202220">of</span> <span m="1202300">these</span> <span m="1202500">constants.</span> <span m="1204270">I</span> <span m="1204410">could</span> <span m="1204550">have</span> <span m="1204740">Fibonacci's</span> <span m="1205360">recurrence</span> <span m="1206710">where</span> <span m="1207460">f</span> <span m="1207690">of</span> <span m="1207810">0</span> <span m="1208390">was</span> <span m="1208560">10</span> <span m="1209010">and</span> <span m="1209120">f</span> <span m="1209300">of</span> <span m="1209430">1</span> <span m="1209700">was</span> <span m="1209850">20</span> <span m="1210220">if</span> <span m="1210350">I</span> <span m="1210410">wanted</span> <span m="1210770">to,</span> <span m="1212040">and</span> <span m="1212190">then</span> <span m="1212380">the</span> <span m="1212760">recurrence</span> <span m="1213090">would</span> <span m="1213190">be</span> <span m="1213280">the</span> <span m="1213390">same</span> <span m="1213810">afterwards.</span> <span m="1215860">And</span> <span m="1216160">it</span> <span m="1216190">will</span> <span m="1216330">turn</span> <span m="1216570">out</span> <span m="1216720">I</span> <span m="1216780">get</span> <span m="1217020">different</span> <span m="1217630">constants</span> <span m="1218180">here.</span> <span m="1219745">All right?</span> <span m="1220180">But</span> <span m="1220290">otherwise,</span> <span m="1220750">the</span> <span m="1220820">form</span> <span m="1221200">is</span> <span m="1221300">going</span> <span m="1221410">to</span> <span m="1221460">look</span> <span m="1221630">like</span> <span m="1221830">this.</span> </p>
<p><span m="1222880">So</span> <span m="1223030">let's</span> <span m="1223340">see</span> <span m="1223570">how</span> <span m="1223890">to</span> <span m="1223990">make</span> <span m="1224230">that</span> <span m="1224420">work.</span> <span m="1224840">Let's</span> <span m="1224900">see</span> <span m="1224990">how</span> <span m="1225110">to</span> <span m="1225170">determine</span> <span m="1225530">the</span> <span m="1225610">constant</span> <span m="1226010">factors.</span> <span m="1248770">OK.</span> <span m="1249390">So</span> <span m="1251320">to</span> <span m="1251740">determine</span> <span m="1252110">the</span> <span m="1252190">constant</span> <span m="1252690">factors,</span> <span m="1265360">we</span> <span m="1265480">plug in</span> <span m="1265900">the</span> <span m="1265990">boundary</span> <span m="1266420">conditions.</span> <span m="1268180">So</span> <span m="1268520">we</span> <span m="1268650">have</span> <span m="1269950">f</span> <span m="1270200">of</span> <span m="1270310">0</span> <span m="1271220">equals</span> <span m="1271720">0</span> <span m="1272520">from</span> <span m="1273020">the</span> <span m="1273190">boundary</span> <span m="1273560">condition.</span> <span m="1274590">And</span> <span m="1274750">now</span> <span m="1274960">we</span> <span m="1275090">plug</span> <span m="1275520">that</span> <span m="1275730">into</span> <span m="1275970">our</span> <span m="1276100">formula</span> <span m="1276660">over</span> <span m="1276900">there.</span> <span m="1280050">That's</span> <span m="1280400">c1</span> <span m="1280970">times</span> <span m="1281740">alpha</span> <span m="1282030">1</span> <span m="1282230">to</span> <span m="1282270">the</span> <span m="1282390">0</span> <span m="1283290">plus</span> <span m="1283950">c2</span> <span m="1284990">alpha</span> <span m="1285310">2</span> <span m="1285550">to</span> <span m="1285610">the</span> <span m="1285720">0.</span> <span m="1286740">Of</span> <span m="1287030">course,</span> <span m="1287790">anything</span> <span m="1288130">to the</span> <span m="1288300">0</span> <span m="1288760">is</span> <span m="1288840">just</span> <span m="1289120">1.</span> <span m="1292480">And</span> <span m="1292670">that</span> <span m="1292900">means</span> <span m="1293270">that</span> <span m="1294470">c2</span> <span m="1295220">equals</span> <span m="1295620">minus</span> <span m="1296100">c1.</span> <span m="1298471">All right?</span> </p>
<p><span m="1299902">And</span> <span m="1300380">now</span> <span m="1300550">I'll</span> <span m="1300640">use</span> <span m="1300990">the</span> <span m="1301080">next</span> <span m="1301450">boundary</span> <span m="1301780">condition</span> <span m="1302280">to</span> <span m="1303420">nail</span> <span m="1303700">them</span> <span m="1303800">down.</span> <span m="1310774">All right?</span> <span m="1311250">So</span> <span m="1311400">I</span> <span m="1311460">know</span> <span m="1311670">also</span> <span m="1312030">that</span> <span m="1312240">f(1)</span> <span m="1313900">is</span> <span m="1314170">1,</span> <span m="1316020">and</span> <span m="1316130">that</span> <span m="1316350">equals</span> <span m="1316760">c1</span> <span m="1318270">1</span> <span m="1318490">plus</span> <span m="1318810">square</span> <span m="1319040">root</span> <span m="1319140">of</span> <span m="1319230">5</span> <span m="1320190">over</span> <span m="1320430">2</span> <span m="1320800">to</span> <span m="1320880">the</span> <span m="1321000">first</span> <span m="1321310">power</span> <span m="1322380">plus</span> <span m="1322660">c2</span> <span m="1323900">1</span> <span m="1324100">minus</span> <span m="1324540">the</span> <span m="1324620">square</span> <span m="1324860">root</span> <span m="1324940">of</span> <span m="1325030">5</span> <span m="1326810">over</span> <span m="1327160">2</span> <span m="1327590">to</span> <span m="1327650">the</span> <span m="1327750">first</span> <span m="1328050">power.</span> <span m="1330050">I</span> <span m="1330150">plug in</span> <span m="1330650">c2</span> <span m="1331170">as</span> <span m="1331300">minus</span> <span m="1331750">c1.</span> <span m="1333720">So</span> <span m="1333820">I</span> <span m="1333890">get</span> <span m="1334080">c1</span> <span m="1334750">1</span> <span m="1334950">plus</span> <span m="1335240">square</span> <span m="1335520">root</span> <span m="1335600">of</span> <span m="1335700">5</span> <span m="1336620">over</span> <span m="1336760">2</span> <span m="1337650">minus</span> <span m="1338100">c1</span> <span m="1339155">1</span> <span m="1339480">minus</span> <span m="1339850">square</span> <span m="1340020">root</span> <span m="1340120">of</span> <span m="1340230">5</span> <span m="1341360">over</span> <span m="1341590">2.</span> </p>
<p><span m="1343790">And now</span> <span m="1344070">that</span> <span m="1344190">I</span> <span m="1344260">can</span> <span m="1344850">factor</span> <span m="1345260">out</span> <span m="1345390">the</span> <span m="1345460">c1,</span> <span m="1347250">I</span> <span m="1347350">get</span> <span m="1347510">1</span> <span m="1347890">minus</span> <span m="1348360">1</span> <span m="1349470">square</span> <span m="1349850">root</span> <span m="1349970">of</span> <span m="1350060">5</span> <span m="1350830">minus</span> <span m="1351530">minus</span> <span m="1351970">square</span> <span m="1352150">root</span> <span m="1352240">of</span> <span m="1352320">5,</span> <span m="1353890">which</span> <span m="1354040">gives</span> <span m="1354200">me</span> <span m="1354320">c1</span> <span m="1355840">2</span> <span m="1356130">square</span> <span m="1356350">root</span> <span m="1356510">of</span> <span m="1356610">5</span> <span m="1357670">over</span> <span m="1357930">2.</span> <span m="1360370">These</span> <span m="1360660">cancel.</span> <span m="1362790">This</span> <span m="1362960">was</span> <span m="1363080">all</span> <span m="1363310">equal</span> <span m="1363530">to</span> <span m="1363600">1,</span> <span m="1364090">so</span> <span m="1364220">that</span> <span m="1364490">means</span> <span m="1364850">that</span> <span m="1365660">c1</span> <span m="1366440">equals</span> <span m="1366970">1</span> <span m="1367480">over</span> <span m="1367760">the</span> <span m="1367890">square</span> <span m="1368180">root</span> <span m="1368280">of</span> <span m="1368360">5.</span> <span m="1370370">And</span> <span m="1370520">of</span> <span m="1370590">course,</span> <span m="1370930">c2</span> <span m="1371380">is</span> <span m="1371480">minus</span> <span m="1371900">that.</span> <span m="1373840">c2</span> <span m="1374760">is</span> <span m="1375480">minus</span> <span m="1376180">1</span> <span m="1376440">over</span> <span m="1376650">the</span> <span m="1376780">square</span> <span m="1376990">root</span> <span m="1377160">of</span> <span m="1377250">5.</span> <span m="1380560">Any</span> <span m="1380710">questions</span> <span m="1382250">so</span> <span m="1382490">far?</span> <span m="1385585">All</span> <span m="1386074">right.</span> </p>
<p><span m="1392930">Now</span> <span m="1392990">I</span> <span m="1393070">could</span> <span m="1393210">write</span> <span m="1393430">out</span> <span m="1393850">the</span> <span m="1394390">formula</span> <span m="1395660">for</span> <span m="1396100">the</span> <span m="1396240">Fibonacci</span> <span m="1396495">numbers.</span> <span m="1402630">All right?</span> <span m="1403130">So</span> <span m="1403190">the</span> <span m="1403340">solution</span> <span m="1403850">is</span> <span m="1408650">f</span> <span m="1408860">of</span> <span m="1408970">n</span> <span m="1410660">equals</span> <span m="1411070">c1,</span> <span m="1412220">which</span> <span m="1412380">is</span> <span m="1412480">1</span> <span m="1412690">over</span> <span m="1412880">square</span> <span m="1413100">root</span> <span m="1413180">of</span> <span m="1413270">5,</span> <span m="1414660">times</span> <span m="1415100">the</span> <span m="1415620">n-th</span> <span m="1415900">power</span> <span m="1416810">of</span> <span m="1416910">the</span> <span m="1416990">first</span> <span m="1417310">root</span> <span m="1420800">plus</span> <span m="1421430">c2,</span> <span m="1422020">which</span> <span m="1422130">is</span> <span m="1422230">minus</span> <span m="1422630">1</span> <span m="1422800">over</span> <span m="1422940">square</span> <span m="1423140">root</span> <span m="1423220">of</span> <span m="1423300">5,</span> <span m="1424140">times</span> <span m="1424450">the</span> <span m="1424570">n-th</span> <span m="1424770">power</span> <span m="1425120">of</span> <span m="1425470">the</span> <span m="1425570">second</span> <span m="1425800">root.</span> <span m="1431030">And</span> <span m="1431630">that</span> <span m="1431900">is</span> <span m="1432470">the</span> <span m="1432590">formula</span> <span m="1433380">for</span> <span m="1433530">the</span> <span m="1433720">n-th</span> <span m="1434010">Fibonacci</span> <span m="1434580">number.</span> </p>
<p><span m="1437580">You</span> <span m="1437700">wouldn't have</span> <span m="1437970">guessed</span> <span m="1438310">that</span> <span m="1438520">to</span> <span m="1438600">start</span> <span m="1438920">with</span> <span m="1439080">obviously.</span> <span m="1440590">That</span> <span m="1440760">would</span> <span m="1440850">require</span> <span m="1441240">pretty</span> <span m="1441490">divine</span> <span m="1441850">inspiration.</span> <span m="1442900">And</span> <span m="1443050">you</span> <span m="1443120">can</span> <span m="1443250">sort</span> <span m="1443440">of</span> <span m="1443510">see</span> <span m="1443740">why</span> <span m="1443930">it</span> <span m="1444030">took</span> <span m="1444150">them</span> <span m="1444310">600</span> <span m="1444890">years</span> <span m="1445640">in</span> <span m="1445790">Europe</span> <span m="1446140">to</span> <span m="1446230">figure</span> <span m="1446560">out</span> <span m="1446710">the</span> <span m="1446870">answer.</span> <span m="1447670">All</span> <span m="1447990">right?</span> <span m="1448160">It's</span> <span m="1448340">not</span> <span m="1448580">the</span> <span m="1448660">first</span> <span m="1448910">thing</span> <span m="1449020">you'd</span> <span m="1449110">think</span> <span m="1449330">about.</span> <span m="1449610">In</span> <span m="1449680">fact,</span> <span m="1449940">if</span> <span m="1450000">somebody</span> <span m="1450330">told</span> <span m="1450630">you</span> <span m="1450730">the</span> <span m="1450890">answer</span> <span m="1451640">and</span> <span m="1451660">said</span> <span m="1451840">this</span> <span m="1452000">is</span> <span m="1452130">it,</span> <span m="1452240">you'd</span> <span m="1452360">go,</span> <span m="1453000">oh,</span> <span m="1453180">give</span> <span m="1453310">me</span> <span m="1453430">a</span> <span m="1453480">break.</span> <span m="1454295">All right?</span> </p>
<p><span m="1454730">It</span> <span m="1454810">does</span> <span m="1454950">not</span> <span m="1455190">look like--</span> <span m="1455420">I</span> <span m="1455760">mean,</span> <span m="1456410">what</span> <span m="1456610">are</span> <span m="1456630">the</span> <span m="1456750">chances</span> <span m="1457210">that</span> <span m="1457450">evaluates</span> <span m="1458170">to</span> <span m="1458290">an</span> <span m="1458380">integer?</span> <span m="1460375">All right?</span> <span m="1460850">It's</span> <span m="1461310">got</span> <span m="1461670">square</span> <span m="1461930">root</span> <span m="1462030">of</span> <span m="1462110">5's</span> <span m="1462580">all</span> <span m="1462800">over</span> <span m="1462980">the</span> <span m="1463100">place.</span> <span m="1465080">Right?</span> <span m="1466140">And</span> <span m="1466300">here</span> <span m="1466520">I'm</span> <span m="1466800">telling</span> <span m="1467060">you</span> <span m="1467410">that</span> <span m="1468400">f</span> <span m="1468640">of</span> <span m="1468760">6,</span> <span m="1470470">the</span> <span m="1470570">sixth</span> <span m="1470990">Fibonacci</span> <span m="1471450">number,</span> <span m="1471620">which</span> <span m="1471810">is</span> <span m="1472000">8,</span> <span m="1472892">3 plus</span> <span m="1473340">5,</span> <span m="1474240">I'm</span> <span m="1474410">telling</span> <span m="1474710">you</span> <span m="1474820">that</span> <span m="1475070">is</span> <span m="1475180">equal</span> <span m="1475480">to</span> <span m="1475580">1</span> <span m="1475820">over</span> <span m="1476030">square</span> <span m="1476210">root</span> <span m="1476360">of</span> <span m="1476430">5</span> <span m="1477760">1</span> <span m="1478070">plus</span> <span m="1478350">the</span> <span m="1478420">square</span> <span m="1478680">root</span> <span m="1478750">of</span> <span m="1478830">5</span> <span m="1479250">over</span> <span m="1479510">2</span> <span m="1479850">to</span> <span m="1479910">the</span> <span m="1480050">sixth</span> <span m="1480440">power</span> <span m="1481550">minus</span> <span m="1482030">1</span> <span m="1482250">over</span> <span m="1482410">square</span> <span m="1482650">root</span> <span m="1482730">of</span> <span m="1482830">5</span> <span m="1484200">1</span> <span m="1484430">minus</span> <span m="1484820">the</span> <span m="1484890">square</span> <span m="1485050">root</span> <span m="1485120">of</span> <span m="1485230">5</span> <span m="1485520">over</span> <span m="1485740">2</span> <span m="1486290">to</span> <span m="1486360">the</span> <span m="1486460">sixth</span> <span m="1486780">power.</span> </p>
<p><span m="1488350">I</span> <span m="1488450">mean,</span> <span m="1488460">would</span> <span m="1488560">you</span> <span m="1488700">believe</span> <span m="1489070">me</span> <span m="1489260">if</span> <span m="1489670">I</span> <span m="1489780">told</span> <span m="1490030">you</span> <span m="1490090">that?</span> <span m="1491350">Probably</span> <span m="1491740">not.</span> <span m="1492970">What</span> <span m="1493130">are</span> <span m="1493150">the</span> <span m="1493270">chances</span> <span m="1493710">that's</span> <span m="1493940">true?</span> <span m="1494560">Somehow,</span> <span m="1495110">magically,</span> <span m="1495760">all</span> <span m="1495940">those</span> <span m="1496110">square</span> <span m="1496340">root</span> <span m="1496440">of</span> <span m="1496520">5's</span> <span m="1496910">all</span> <span m="1497080">go</span> <span m="1497190">away,</span> <span m="1498450">and</span> <span m="1498560">you're</span> <span m="1498640">just</span> <span m="1498850">left</span> <span m="1499020">with</span> <span m="1499150">8,</span> <span m="1500052">a</span> <span m="1500400">simple</span> <span m="1500700">integer.</span> <span m="1503130">OK?</span> <span m="1506130">All</span> <span m="1506250">right.</span> </p>
<p><span m="1508180">Yeah, there's</span> <span m="1508470">sort</span> <span m="1508600">of</span> <span m="1508660">some</span> <span m="1508840">more</span> <span m="1510610">interesting</span> <span m="1511010">things</span> <span m="1511290">about</span> <span m="1511580">this.</span> <span m="1513895">What</span> <span m="1514360">happens</span> <span m="1514810">to</span> <span m="1514910">this</span> <span m="1515340">value</span> <span m="1516380">here</span> <span m="1516870">as</span> <span m="1517030">n</span> <span m="1517280">gets</span> <span m="1517530">large?</span> <span m="1520450">What</span> <span m="1520550">does</span> <span m="1520680">it</span> <span m="1520840">do</span> <span m="1521060">as</span> <span m="1521170">n</span> <span m="1521320">gets</span> <span m="1521540">large?</span> </p>
<p><span m="1522938">AUDIENCE: Gets</span> <span m="1523412">small.</span> </p>
<p><span m="1526260">PROFESSOR: It gets</span> <span m="1526480">small</span> <span m="1528170">Because</span> <span m="1528480">1</span> <span m="1528810">minus</span> <span m="1529220">the</span> <span m="1529300">square</span> <span m="1529590">root</span> <span m="1529690">of</span> <span m="1529780">5</span> <span m="1530210">over</span> <span m="1530470">2</span> <span m="1530930">is</span> <span m="1531040">about</span> <span m="1531500">0.6.</span> <span m="1534480">This</span> <span m="1534670">is</span> <span m="1534750">about,</span> <span m="1535320">here,</span> <span m="1535650">this</span> <span m="1535870">is</span> <span m="1535930">about</span> <span m="1536450">0.618</span> <span m="1537620">something</span> <span m="1537980">or</span> <span m="1538040">other.</span> <span m="1539120">And</span> <span m="1539250">if</span> <span m="1539330">I</span> <span m="1539440">take</span> <span m="1539680">a</span> <span m="1539750">fraction</span> <span m="1540770">less</span> <span m="1540970">than</span> <span m="1541090">1</span> <span m="1541410">to the n-th</span> <span m="1541690">power,</span> <span m="1542050">it</span> <span m="1542110">goes</span> <span m="1542340">to</span> <span m="1542420">0.</span> <span m="1544440">It</span> <span m="1544560">goes</span> <span m="1544820">away.</span> <span m="1546430">So</span> <span m="1546550">in</span> <span m="1546630">fact,</span> <span m="1547010">as</span> <span m="1547190">n</span> <span m="1547470">gets</span> <span m="1547730">large,</span> <span m="1549830">this</span> <span m="1550140">is</span> <span m="1550340">what</span> <span m="1550650">the</span> <span m="1550780">n-th</span> <span m="1550970">Fibonacci</span> <span m="1551500">number</span> <span m="1551760">starts</span> <span m="1552050">to</span> <span m="1552140">look</span> <span m="1552360">like.</span> </p>
<p><span m="1554050">In</span> <span m="1554110">particular,</span> <span m="1554680">f</span> <span m="1554890">of</span> <span m="1554990">n</span> <span m="1556260">equals</span> <span m="1556750">just</span> <span m="1557050">that</span> <span m="1557270">first</span> <span m="1557610">one--</span> <span m="1558010">1</span> <span m="1558250">over</span> <span m="1558390">square</span> <span m="1558590">root</span> <span m="1558690">of</span> <span m="1558790">5</span> <span m="1559660">times</span> <span m="1560080">the</span> <span m="1560160">golden</span> <span m="1560530">ratio</span> <span m="1564950">to</span> <span m="1565050">the</span> <span m="1565130">n-th</span> <span m="1565270">power</span> <span m="1565530">plus</span> <span m="1565920">some</span> <span m="1566260">error</span> <span m="1566560">term,</span> <span m="1566860">delta</span> <span m="1567200">n.</span> <span m="1569260">And</span> <span m="1569550">delta</span> <span m="1569900">n</span> <span m="1570250">is</span> <span m="1573330">less</span> <span m="1573640">than</span> <span m="1573970">a</span> <span m="1574050">1/10</span> <span m="1576111">for</span> <span m="1576600">n</span> <span m="1576930">greater</span> <span m="1577170">than</span> <span m="1577330">4.</span> <span m="1580300">And</span> <span m="1580690">it</span> <span m="1581020">is</span> <span m="1581210">little l</span> <span m="1581590">of</span> <span m="1581690">1</span> <span m="1581950">in</span> <span m="1582030">general</span> <span m="1584510">going</span> <span m="1584760">to</span> <span m="1584800">0.</span> <span m="1587330">So</span> <span m="1587460">in</span> <span m="1587550">fact,</span> <span m="1587820">the</span> <span m="1587950">n-th</span> <span m="1588140">Fibonacci</span> <span m="1588640">number</span> <span m="1589070">is</span> <span m="1589160">about</span> <span m="1589600">the</span> <span m="1589750">n-th</span> <span m="1590000">power of</span> <span m="1590420">the</span> <span m="1590540">golden</span> <span m="1590880">ratio</span> <span m="1591350">divided</span> <span m="1591720">by</span> <span m="1591830">square</span> <span m="1592130">root of</span> <span m="1592280">5.</span> <span m="1595020">Yeah?</span> </p>
<p><span m="1595922">AUDIENCE: [INAUDIBLE]</span> <span m="1596786">top of</span> <span m="1597218">the</span> <span m="1597650">right</span> <span m="1598082">board--</span> </p>
<p><span m="1598520">PROFESSOR: Yeah?</span> </p>
<p><span m="1598730">AUDIENCE: --you say that</span> <span m="1600698">you can</span> <span m="1601190">pick sort of</span> <span m="1601682">any constants</span> <span m="1602174">that</span> <span m="1603158">satisfy</span> <span m="1603650">it?</span> </p>
<p><span m="1604142">PROFESSOR: Yep.</span> </p>
<p><span m="1604634">AUDIENCE: Does that</span> <span m="1605126">shift</span> <span m="1605618">the</span> <span m="1606110">sequence</span> <span m="1606602">over,</span> <span m="1607094">or what</span> <span m="1607586">does that</span> <span m="1608078">do?</span> <span m="1608570">I guess I don't</span> <span m="1609062">understand.</span> </p>
<p><span m="1610046">PROFESSOR: OK.</span> <span m="1611810">You're</span> <span m="1612000">asking,</span> <span m="1613630">what</span> <span m="1613720">are</span> <span m="1613740">these</span> <span m="1613970">things</span> <span m="1614180">doing</span> <span m="1614430">here?</span> </p>
<p><span m="1615796">AUDIENCE: Yeah.</span> <span m="1616234">And</span> <span m="1616672">why</span> <span m="1617110">[? isn't there-- ?]</span> <span m="1617550">why are they</span> <span m="1617915">also</span> <span m="1618280">solutions?</span> <span m="1618570">Because this</span> <span m="1619000">is the</span> <span m="1619430">right</span> <span m="1619860">ones?</span> </p>
<p><span m="1620125">PROFESSOR: The</span> <span m="1620390">right</span> <span m="1620680">one.</span> <span m="1620890">OK.</span> <span m="1621760">So</span> <span m="1622960">there's</span> <span m="1623230">two</span> <span m="1623400">parts</span> <span m="1623770">to</span> <span m="1623910">a</span> <span m="1624000">recurrence.</span> <span m="1625120">There's</span> <span m="1627460">this</span> <span m="1627740">part,</span> <span m="1629110">OK,</span> <span m="1630230">and</span> <span m="1630640">there's</span> <span m="1631120">the</span> <span m="1632120">boundary</span> <span m="1632510">conditions--</span> <span m="1634160">f(0)</span> <span m="1634450">equals</span> <span m="1634850">0</span> <span m="1635190">and</span> <span m="1635260">f(1)</span> <span m="1635610">equals</span> <span m="1635880">1.</span> <span m="1637250">There are</span> <span m="1637630">really</span> <span m="1637850">two</span> <span m="1638020">parts</span> <span m="1638390">to</span> <span m="1638550">it.</span> <span m="1639450">If</span> <span m="1639590">you</span> <span m="1639750">change</span> <span m="1640430">the</span> <span m="1640510">boundary</span> <span m="1640890">conditions,</span> <span m="1641610">you</span> <span m="1641880">would</span> <span m="1642110">change</span> <span m="1642570">the</span> <span m="1642650">rest</span> <span m="1642900">of</span> <span m="1642940">the</span> <span m="1643030">terms,</span> <span m="1644900">right?</span> <span m="1646240">If</span> <span m="1646380">I</span> <span m="1646520">started</span> <span m="1646810">with</span> <span m="1647060">10 and</span> <span m="1647340">20,</span> <span m="1647713">f of</span> <span m="1648086">2</span> <span m="1648460">would</span> <span m="1648620">be</span> <span m="1648770">30,</span> <span m="1650150">OK,</span> <span m="1650880">if</span> <span m="1651030">I'm</span> <span m="1651150">using</span> <span m="1653000">this</span> <span m="1653460">recurrence</span> <span m="1653930">form</span> <span m="1654180">for</span> <span m="1654330">n</span> <span m="1654510">bigger than</span> <span m="1654750">or</span> <span m="1654840">equal</span> <span m="1655040">to</span> <span m="1655130">2.</span> </p>
<p><span m="1656522">Now,</span> <span m="1656980">so</span> <span m="1657120">when</span> <span m="1657230">I'm</span> <span m="1657360">computing</span> <span m="1657800">a</span> <span m="1657860">solution,</span> <span m="1658990">first</span> <span m="1659450">I'm</span> <span m="1659580">saying,</span> <span m="1660200">let's</span> <span m="1660420">ignore</span> <span m="1660730">the</span> <span m="1660840">boundary</span> <span m="1661250">conditions</span> <span m="1661860">and</span> <span m="1661970">look</span> <span m="1662130">at</span> <span m="1662200">all</span> <span m="1662580">the</span> <span m="1662690">recurrences</span> <span m="1663320">that</span> <span m="1663430">share</span> <span m="1663720">this</span> <span m="1663920">part.</span> <span m="1665770">And</span> <span m="1666000">that</span> <span m="1666270">family</span> <span m="1666760">of</span> <span m="1666850">recurrences,</span> <span m="1668470">I'm</span> <span m="1668650">claiming,</span> <span m="1669960">have</span> <span m="1670260">these</span> <span m="1670540">solutions.</span> <span m="1671580">This</span> <span m="1671780">is</span> <span m="1671890">the</span> <span m="1672000">solution</span> <span m="1672480">space</span> <span m="1673310">if</span> <span m="1673470">I</span> <span m="1673550">just</span> <span m="1674130">worry</span> <span m="1674450">about</span> <span m="1675850">this</span> <span m="1676060">part</span> <span m="1676410">and</span> <span m="1676550">not</span> <span m="1676930">the</span> <span m="1677140">boundary</span> <span m="1677500">conditions.</span> </p>
<p><span m="1679460">Now,</span> <span m="1680540">once</span> <span m="1680870">I</span> <span m="1680940">plug</span> <span m="1681250">the</span> <span m="1681330">boundary</span> <span m="1681700">conditions</span> <span m="1682200">in,</span> <span m="1683160">that</span> <span m="1683440">determines</span> <span m="1684860">which</span> <span m="1685260">values</span> <span m="1685650">of</span> <span m="1685730">c1 and</span> <span m="1686220">c2</span> <span m="1686640">I</span> <span m="1686720">get.</span> <span m="1688220">And</span> <span m="1688440">so</span> <span m="1688950">if</span> <span m="1689110">I</span> <span m="1689240">take</span> <span m="1689470">the</span> <span m="1689540">version</span> <span m="1689950">of</span> <span m="1690040">the</span> <span m="1690120">recurrence</span> <span m="1690630">where</span> <span m="1690790">f(0)</span> <span m="1691740">is</span> <span m="1691840">0</span> <span m="1692100">and</span> <span m="1692180">f(1)</span> <span m="1692590">is</span> <span m="1692700">1,</span> <span m="1694050">then</span> <span m="1694440">c1</span> <span m="1694990">is</span> <span m="1695100">going</span> <span m="1695200">to</span> <span m="1695240">be</span> <span m="1695330">1</span> <span m="1695610">over</span> <span m="1695800">root</span> <span m="1696000">5,</span> <span m="1696400">and</span> <span m="1696510">c2</span> <span m="1696900">is</span> <span m="1697000">going</span> <span m="1697100">to</span> <span m="1697140">be</span> <span m="1697250">negative</span> <span m="1697890">1</span> <span m="1698080">over</span> <span m="1698240">root</span> <span m="1698420">5.</span> </p>
<p><span m="1699720">If</span> <span m="1700010">I</span> <span m="1700450">use</span> <span m="1701030">this</span> <span m="1701750">recurrence</span> <span m="1702370">with</span> <span m="1703160">f(0)</span> <span m="1703660">equals</span> <span m="1703930">10</span> <span m="1704300">and</span> <span m="1704390">f(1)</span> <span m="1705190">equals</span> <span m="1705480">20,</span> <span m="1706120">I</span> <span m="1706240">get</span> <span m="1706430">different</span> <span m="1706700">constants,</span> <span m="1708430">but</span> <span m="1708930">these</span> <span m="1709160">powers</span> <span m="1709710">stay</span> <span m="1709950">the</span> <span m="1710060">same.</span> <span m="1711690">In fact,</span> <span m="1711880">maybe</span> <span m="1712030">we'll</span> <span m="1712170">make</span> <span m="1712360">an</span> <span m="1712450">exercise</span> <span m="1712970">like</span> <span m="1713170">that from</span> <span m="1713470">the</span> <span m="1713630">problem</span> <span m="1713970">set</span> <span m="1714170">to</span> <span m="1714230">figure</span> <span m="1714550">out</span> <span m="1714660">what</span> <span m="1714790">it</span> <span m="1714890">is</span> <span m="1715160">if</span> <span m="1715250">I</span> <span m="1715320">started</span> <span m="1715570">with</span> <span m="1715690">10</span> <span m="1715870">and</span> <span m="1715960">20.</span> <span m="1717450">OK?</span> <span m="1718020">So</span> <span m="1718160">that's</span> <span m="1718450">what</span> <span m="1718670">the</span> <span m="1718750">constants</span> <span m="1720030">have</span> <span m="1720240">to</span> <span m="1720340">do--</span> <span m="1720910">they</span> <span m="1721300">come</span> <span m="1721480">from</span> <span m="1721540">the</span> <span m="1721640">boundary</span> <span m="1721990">condition,</span> <span m="1722990">which</span> <span m="1723160">is</span> <span m="1723240">sort</span> <span m="1723390">of</span> <span m="1723480">a</span> <span m="1723550">cool</span> <span m="1723820">fact.</span> <span m="1725591">Does</span> <span m="1726080">that</span> <span m="1726230">make</span> <span m="1726420">sense?</span> </p>
<p><span m="1727280">AUDIENCE: Yeah.</span> </p>
<p><span m="1728060">PROFESSOR: Any</span> <span m="1728160">other</span> <span m="1728400">questions</span> <span m="1728890">about</span> <span m="1729960">what</span> <span m="1730140">I</span> <span m="1730190">was</span> <span m="1730410">doing?</span> <span m="1731570">So</span> <span m="1731680">we</span> <span m="1731780">used</span> <span m="1732130">guess</span> <span m="1732450">and</span> <span m="1732560">verify,</span> <span m="1733290">but</span> <span m="1734510">I</span> <span m="1734630">sort</span> <span m="1734800">of</span> <span m="1734910">guessed</span> <span m="1735190">a</span> <span m="1735340">form</span> <span m="1736050">first.</span> <span m="1736590">And</span> <span m="1736740">as</span> <span m="1736900">I</span> <span m="1736980">verified,</span> <span m="1738240">I</span> <span m="1738340">sort</span> <span m="1738540">of</span> <span m="1738880">revised</span> <span m="1739490">my</span> <span m="1739650">guess</span> <span m="1740030">along</span> <span m="1740350">the</span> <span m="1740450">way.</span> <span m="1741660">All</span> <span m="1741730">right?</span> <span m="1741890">So I</span> <span m="1742040">started</span> <span m="1742370">guessing</span> <span m="1742750">this.</span> <span m="1743160">And then</span> <span m="1743360">I</span> <span m="1743410">said,</span> <span m="1743670">well,</span> <span m="1744660">let's</span> <span m="1744850">refine</span> <span m="1745290">that</span> <span m="1745550">guess</span> <span m="1746030">so</span> <span m="1746040">that</span> <span m="1746820">really</span> <span m="1747210">it's</span> <span m="1749301">one</span> <span m="1749720">of</span> <span m="1749790">these</span> <span m="1750000">two</span> <span m="1750140">guys.</span> </p>
<p><span m="1751360">And</span> <span m="1751500">then</span> <span m="1751660">I</span> <span m="1751760">said--</span> <span m="1752080">I</span> <span m="1752140">used</span> <span m="1752350">a</span> <span m="1752440">fact</span> <span m="1752710">that I didn't</span> <span m="1752980">prove--</span> <span m="1753285">that</span> <span m="1753590">said</span> <span m="1753710">I</span> <span m="1753780">could</span> <span m="1753940">use</span> <span m="1754120">really</span> <span m="1754410">any</span> <span m="1754600">linear</span> <span m="1754890">combination,</span> <span m="1755750">plugged</span> <span m="1756670">it</span> <span m="1756730">into</span> <span m="1756960">the</span> <span m="1757030">base</span> <span m="1757280">cases,</span> <span m="1757570">and</span> <span m="1757720">got</span> <span m="1757880">my</span> <span m="1757970">constants.</span> <span m="1760640">All right?</span> <span m="1760800">Now,</span> <span m="1760940">this</span> <span m="1761130">will</span> <span m="1761260">lead</span> <span m="1761480">up</span> <span m="1761600">to</span> <span m="1761720">a</span> <span m="1761800">formula</span> <span m="1762540">or</span> <span m="1762700">an</span> <span m="1762760">approach</span> <span m="1763870">so</span> <span m="1763990">you</span> <span m="1764030">don't</span> <span m="1764180">have</span> <span m="1764290">to</span> <span m="1764380">go</span> <span m="1764540">through</span> <span m="1764650">this</span> <span m="1764850">on</span> <span m="1764950">your</span> <span m="1765140">own.</span> <span m="1765310">You</span> <span m="1765370">just</span> <span m="1765600">plug in</span> <span m="1765900">the</span> <span m="1765980">formula</span> <span m="1767110">in</span> <span m="1767250">general.</span> </p>
<p><span m="1770330">OK.</span> <span m="1773500">Let's</span> <span m="1773710">get</span> <span m="1773840">back</span> <span m="1774020">to</span> <span m="1774080">the</span> <span m="1774200">original</span> <span m="1774540">question</span> <span m="1775000">of</span> <span m="1775090">when</span> <span m="1775360">all</span> <span m="1775600">the</span> <span m="1775700">jobs</span> <span m="1776100">are</span> <span m="1776170">filled.</span> <span m="1777660">For</span> <span m="1777830">what</span> <span m="1778140">value</span> <span m="1778590">of</span> <span m="1778680">n</span> <span m="1780040">is</span> <span m="1780320">f</span> <span m="1780550">of</span> <span m="1780700">n</span> <span m="1780910">bigger</span> <span m="1781330">than</span> <span m="1781520">M?</span> <span m="1783320">All right?</span> <span m="1788790">Let's</span> <span m="1788980">do</span> <span m="1789120">that.</span> <span m="1803940">OK.</span> <span m="1804450">To</span> <span m="1804850">see</span> <span m="1805120">when</span> <span m="1805250">all</span> <span m="1805480">the</span> <span m="1805580">jobs</span> <span m="1806010">are</span> <span m="1806080">filled,</span> <span m="1809310">all</span> <span m="1809590">M</span> <span m="1809820">jobs,</span> <span m="1820130">for the</span> <span m="1820390">n</span> <span m="1820770">when</span> <span m="1821100">f of</span> <span m="1821370">n</span> <span m="1822360">is</span> <span m="1822550">bigger</span> <span m="1822860">and</span> <span m="1822960">equal</span> <span m="1823190">to</span> <span m="1823720">M,</span> <span m="1824876">when</span> <span m="1825300">we</span> <span m="1825510">got</span> <span m="1825690">a</span> <span m="1825740">Fibonacci</span> <span m="1826320">number</span> <span m="1826510">that's</span> <span m="1826700">bigger</span> <span m="1826890">than</span> <span m="1827020">M.</span> </p>
<p><span m="1828300">And</span> <span m="1828490">we</span> <span m="1828610">just</span> <span m="1828870">showed</span> <span m="1829420">that</span> <span m="1829610">basically</span> <span m="1829910">f</span> <span m="1830210">of</span> <span m="1830360">n,</span> <span m="1831770">well,</span> <span m="1831950">it</span> <span m="1832000">equals</span> <span m="1832340">this</span> <span m="1832770">where</span> <span m="1832860">that's</span> <span m="1833070">a</span> <span m="1833150">tiny</span> <span m="1833530">thing.</span> <span m="1835270">All</span> <span m="1835360">right?</span> <span m="1835590">So</span> <span m="1835660">we</span> <span m="1835700">need</span> <span m="1835830">to</span> <span m="1835890">figure</span> <span m="1836240">out</span> <span m="1836480">when</span> <span m="1836730">is</span> <span m="1837550">1</span> <span m="1837740">over</span> <span m="1837930">square</span> <span m="1838160">root of</span> <span m="1838300">5</span> <span m="1839580">times</span> <span m="1840020">the</span> <span m="1840120">golden</span> <span m="1840410">ratio</span> <span m="1840820">to</span> <span m="1840880">the</span> <span m="1840980">n-th</span> <span m="1841170">power--</span> <span m="1843580">plus</span> <span m="1844080">I</span> <span m="1844140">have</span> <span m="1844300">this</span> <span m="1844470">tiny</span> <span m="1844800">little</span> <span m="1845010">thing</span> <span m="1845330">that</span> <span m="1845430">doesn't</span> <span m="1845580">matter--</span> <span m="1847430">is</span> <span m="1847650">bigger than</span> <span m="1847780">or</span> <span m="1847900">equal</span> <span m="1848130">to</span> <span m="1848220">M?</span> <span m="1850200">So</span> <span m="1850360">I can</span> <span m="1850540">solve</span> <span m="1851690">for</span> <span m="1851950">n</span> <span m="1852250">now</span> <span m="1852530">by</span> <span m="1852680">subtracting</span> <span m="1853330">the</span> <span m="1853440">delta</span> <span m="1853810">term,</span> <span m="1854790">multiplying</span> <span m="1855370">by</span> <span m="1855480">square</span> <span m="1855720">root</span> <span m="1855810">of</span> <span m="1855920">5.</span> <span m="1857300">That</span> <span m="1857490">gives</span> <span m="1857760">me</span> <span m="1859730">the</span> <span m="1859800">golden</span> <span m="1860140">ratio</span> <span m="1860670">to</span> <span m="1860750">the</span> <span m="1860890">n-th</span> <span m="1861080">power</span> <span m="1862260">bigger</span> <span m="1862500">than</span> <span m="1862690">or</span> <span m="1862800">equal</span> <span m="1863070">to</span> <span m="1863990">square</span> <span m="1864390">root</span> <span m="1864560">5</span> <span m="1865880">times</span> <span m="1866240">M</span> <span m="1866760">minus</span> <span m="1867060">the</span> <span m="1867360">tiny</span> <span m="1867700">thing.</span> </p>
<p><span m="1870800">Now</span> <span m="1871030">I</span> <span m="1871150">take</span> <span m="1871450">logs.</span> <span m="1874040">And</span> <span m="1874630">I'm</span> <span m="1874810">in</span> <span m="1874890">trouble</span> <span m="1875320">when</span> <span m="1875650">n</span> <span m="1876580">is</span> <span m="1876780">bigger</span> <span m="1877050">than</span> <span m="1877290">or</span> <span m="1877410">equal</span> <span m="1877690">to</span> <span m="1880640">log</span> <span m="1881450">of</span> <span m="1881510">this</span> <span m="1888340">over</span> <span m="1889610">log</span> <span m="1890080">of</span> <span m="1890430">the</span> <span m="1890520">base.</span> <span m="1898710">In</span> <span m="1898860">other</span> <span m="1899050">words,</span> <span m="1899680">this</span> <span m="1899840">is</span> <span m="1900290">theta.</span> <span m="1901070">What's</span> <span m="1901280">this</span> <span m="1901550">theta</span> <span m="1901870">of</span> <span m="1902120">here?</span> <span m="1905830">That</span> <span m="1906120">goes</span> <span m="1906360">away.</span> <span m="1908500">I</span> <span m="1908600">can</span> <span m="1908790">skip</span> <span m="1909080">all</span> <span m="1909240">the</span> <span m="1909330">constant</span> <span m="1909780">factors.</span> <span m="1910450">What's</span> <span m="1910670">this</span> <span m="1910910">theta</span> <span m="1911240">of</span> <span m="1911470">here,</span> <span m="1911780">this</span> <span m="1911950">expression,</span> <span m="1912225">in</span> <span m="1912500">terms of</span> <span m="1912830">theta</span> <span m="1913400">of</span> <span m="1913530">something</span> <span m="1913960">to</span> <span m="1914040">do with</span> <span m="1914300">some</span> <span m="1914530">function</span> <span m="1914860">of</span> <span m="1914950">M?</span> </p>
<p><span m="1916224">AUDIENCE: Log of M?</span> </p>
<p><span m="1917220">PROFESSOR: Log</span> <span m="1917540">of</span> <span m="1917640">M.</span> <span m="1920540">So</span> <span m="1920850">the</span> <span m="1920970">jobs</span> <span m="1921410">get</span> <span m="1921620">filled</span> <span m="1921940">in</span> <span m="1922020">log</span> <span m="1922330">of</span> <span m="1922430">M</span> <span m="1922620">years.</span> <span m="1922910">And</span> <span m="1923200">in</span> <span m="1923300">fact,</span> <span m="1923870">if</span> <span m="1923970">I</span> <span m="1924070">plugged</span> <span m="1924335">in,</span> <span m="1924600">for</span> <span m="1924710">example,</span> <span m="1925250">M</span> <span m="1925470">equals</span> <span m="1925760">10,000</span> <span m="1926430">into</span> <span m="1926560">this</span> <span m="1926760">expression,</span> <span m="1927810">I</span> <span m="1927930">would</span> <span m="1928120">find that</span> <span m="1928500">all</span> <span m="1928740">the</span> <span m="1928830">jobs</span> <span m="1929220">are</span> <span m="1929290">filled</span> <span m="1929570">in</span> <span m="1929670">20</span> <span m="1930000">years,</span> <span m="1931600">which</span> <span m="1931800">is</span> <span m="1931900">more</span> <span m="1932120">or</span> <span m="1932140">less</span> <span m="1932400">what</span> <span m="1932540">happened</span> <span m="1932900">in</span> <span m="1932970">computer</span> <span m="1933330">science.</span> <span m="1934780">Ballpark,</span> <span m="1935580">those</span> <span m="1935790">numbers</span> <span m="1936140">are</span> <span m="1936590">roughly</span> <span m="1937010">correct.</span> <span m="1938320">All right?</span> <span m="1938720">But</span> <span m="1938840">you</span> <span m="1938920">can</span> <span m="1939060">solve</span> <span m="1939410">it</span> <span m="1939470">exactly</span> <span m="1940060">now</span> <span m="1940380">by</span> <span m="1940510">just</span> <span m="1940740">plugging</span> <span m="1941040">in</span> <span m="1941110">whatever M</span> <span m="1941530">you</span> <span m="1941690">want.</span> <span m="1942490">That's</span> <span m="1942700">a</span> <span m="1942760">fraction</span> <span m="1943170">less</span> <span m="1943380">than</span> <span m="1943550">1/10.</span> <span m="1943870">And you</span> <span m="1944010">can</span> <span m="1944160">solve</span> <span m="1944530">to</span> <span m="1944610">find</span> <span m="1944890">out</span> <span m="1945020">when</span> <span m="1945190">your</span> <span m="1945330">population</span> <span m="1946110">gets</span> <span m="1946320">that</span> <span m="1946530">big.</span> <span m="1950080">OK.</span> <span m="1950350">Any</span> <span m="1950490">questions?</span> </p>
<p><span m="1956560">OK.</span> <span m="1956910">So</span> <span m="1957040">now</span> <span m="1958150">what</span> <span m="1958270">I</span> <span m="1958320">want</span> <span m="1958520">to</span> <span m="1958580">do</span> <span m="1958830">is</span> <span m="1958900">show</span> <span m="1959160">you</span> <span m="1959250">how</span> <span m="1959380">to</span> <span m="1959450">use</span> <span m="1959620">this</span> <span m="1959720">same</span> <span m="1960090">idea</span> <span m="1961010">to</span> <span m="1961270">solve</span> <span m="1961860">any</span> <span m="1962360">linear</span> <span m="1962690">recurrence,</span> <span m="1963880">and</span> <span m="1963950">this</span> <span m="1964090">is</span> <span m="1964200">the</span> <span m="1964270">process</span> <span m="1964720">you'll</span> <span m="1964900">go</span> <span m="1965070">through</span> <span m="1965330">to</span> <span m="1965460">do  it.</span> <span m="1981625">All</span> <span m="1982140">right. So</span> <span m="1982480">we're going to</span> <span m="1982590">solve</span> <span m="1984890">a</span> <span m="1985010">general</span> <span m="1985780">linear</span> <span m="1986080">recurrence.</span> <span m="2000040">So</span> <span m="2001230">in</span> <span m="2001290">this</span> <span m="2001490">case,</span> <span m="2002700">we've</span> <span m="2002720">got</span> <span m="2004070">f</span> <span m="2004320">of</span> <span m="2004440">n</span> <span m="2005810">is</span> <span m="2006190">the</span> <span m="2006310">sum</span> <span m="2006840">from</span> <span m="2007040">i</span> <span m="2007440">equals</span> <span m="2007810">1</span> <span m="2008100">to</span> <span m="2008240">d,</span> <span m="2008565">for an</span> <span m="2008890">order</span> <span m="2009180">d</span> <span m="2009410">recurrence,</span> <span m="2010590">a</span> <span m="2010800">sub</span> <span m="2010970">i</span> <span m="2011530">f</span> <span m="2011720">of</span> <span m="2011850">n</span> <span m="2012000">minus</span> <span m="2012370">i.</span> </p>
<p><span m="2014530">And</span> <span m="2014960">we've</span> <span m="2015110">got</span> <span m="2015310">to</span> <span m="2015370">have</span> <span m="2015660">boundary</span> <span m="2016170">conditions.</span> <span m="2018280">So</span> <span m="2018565">we'll</span> <span m="2018870">have</span> <span m="2019150">f</span> <span m="2019340">of</span> <span m="2019430">0</span> <span m="2020320">is</span> <span m="2020810">b0,</span> <span m="2022640">f</span> <span m="2022890">of</span> <span m="2023010">1</span> <span m="2024020">is</span> <span m="2024280">b1.</span> <span m="2026570">And</span> <span m="2026840">how</span> <span m="2027030">many</span> <span m="2027230">boundary</span> <span m="2027620">conditions do</span> <span m="2028100">you</span> <span m="2028220">think</span> <span m="2028420">I'm</span> <span m="2028500">going</span> <span m="2028620">to</span> <span m="2028670">need</span> <span m="2028870">to</span> <span m="2028930">have?</span> <span m="2029390">Any</span> <span m="2029560">guesses?</span> <span m="2031790">I</span> <span m="2031920">had</span> <span m="2032180">two</span> <span m="2032440">for</span> <span m="2032570">Fibonacci.</span> <span m="2034242">In</span> <span m="2034620">this</span> <span m="2034840">case,</span> <span m="2035080">I've</span> <span m="2035180">got</span> <span m="2035430">d</span> <span m="2035610">terms.</span> </p>
<p><span m="2036320">AUDIENCE: d.</span> </p>
<p><span m="2037170">PROFESSOR: d.</span> <span m="2037600">I'm going to</span> <span m="2037950">need d</span> <span m="2038230">boundary</span> <span m="2038610">conditions.</span> <span m="2039090">So</span> <span m="2039250">we'll</span> <span m="2039330">go</span> <span m="2039460">all</span> <span m="2039680">the</span> <span m="2039780">way</span> <span m="2040020">to</span> <span m="2041160">f</span> <span m="2041450">of</span> <span m="2041600">d</span> <span m="2041800">minus</span> <span m="2042220">1</span> <span m="2043650">equal</span> <span m="2043920">to</span> <span m="2044270">b</span> <span m="2044480">of</span> <span m="2044570">d</span> <span m="2044730">minus</span> <span m="2045110">1.</span> <span m="2047992">All</span> <span m="2048429">right?</span> <span m="2048780">And</span> <span m="2048920">now</span> <span m="2049060">I'm</span> <span m="2049159">going</span> <span m="2049280">to</span> <span m="2050780">try</span> <span m="2053750">f</span> <span m="2053989">of</span> <span m="2054130">n</span> <span m="2055210">is</span> <span m="2055469">alpha</span> <span m="2055920">to the</span> <span m="2056130">n.</span> <span m="2056839">We're</span> <span m="2057270">going</span> <span m="2057400">to</span> <span m="2057460">plug</span> <span m="2057780">it</span> <span m="2057840">into</span> <span m="2058040">this</span> <span m="2058250">expression.</span> <span m="2060989">And</span> <span m="2061190">then</span> <span m="2061360">when</span> <span m="2061500">we</span> <span m="2061630">do,</span> <span m="2061969">we</span> <span m="2062100">get</span> <span m="2064020">alpha to the</span> <span m="2064389">n</span> <span m="2065090">equals</span> <span m="2066699">a1</span> <span m="2068360">alpha to the</span> <span m="2068690">n</span> <span m="2069050">minus</span> <span m="2069409">1</span> <span m="2070929">plus</span> <span m="2071480">a2</span> <span m="2072750">alpha</span> <span m="2073120">to</span> <span m="2073159">the</span> <span m="2073250">n</span> <span m="2073380">minus</span> <span m="2073739">2</span> <span m="2076080">all</span> <span m="2076310">the</span> <span m="2076370">way</span> <span m="2076469">down</span> <span m="2076760">to</span> <span m="2076920">a</span> <span m="2077130">sub</span> <span m="2077360">d</span> <span m="2081060">alpha</span> <span m="2081325">to the</span> <span m="2081590">n</span> <span m="2081989">minus</span> <span m="2082330">d.</span> <span m="2084314">All right?</span> <span m="2084810">Just</span> <span m="2085020">plugging</span> <span m="2085360">into</span> <span m="2085530">there.</span> </p>
<p><span m="2087280">I</span> <span m="2087440">can</span> <span m="2087610">divide</span> <span m="2088100">everything</span> <span m="2088440">by</span> <span m="2088590">alpha to the</span> <span m="2088909">n</span> <span m="2089210">minus</span> <span m="2089580">d,</span> <span m="2090790">and</span> <span m="2090889">that</span> <span m="2091110">gives</span> <span m="2091370">me</span> <span m="2092429">alpha to the</span> <span m="2092800">d</span> <span m="2093000">equals</span> <span m="2094290">a1</span> <span m="2095393">alpha to the</span> <span m="2095786">d</span> <span m="2096230">minus</span> <span m="2096620">1</span> <span m="2097979">a2</span> <span m="2099440">alpha</span> <span m="2099770">to the</span> <span m="2099980">d</span> <span m="2100250">minus</span> <span m="2100700">2</span> <span m="2103370">ad</span> <span m="2103990">times</span> <span m="2104320">alpha</span> <span m="2104620">to the</span> <span m="2104740">0,</span> <span m="2105630">which</span> <span m="2105840">is</span> <span m="2105940">just</span> <span m="2106230">1.</span> <span m="2108760">And</span> <span m="2108890">I</span> <span m="2108940">can</span> <span m="2109120">rewrite</span> <span m="2109660">this</span> <span m="2109990">as</span> <span m="2110340">a</span> <span m="2111050">polynomial</span> <span m="2112050">equal</span> <span m="2112300">to</span> <span m="2112400">0.</span> <span m="2122800">So</span> <span m="2123300">that</span> <span m="2123530">means</span> <span m="2123870">that</span> <span m="2124990">alpha</span> <span m="2125320">to the</span> <span m="2125600">d</span> <span m="2125990">minus</span> <span m="2126255">a1</span> <span m="2127741">alpha</span> <span m="2128150">to the</span> <span m="2128300">d</span> <span m="2128380">minus</span> <span m="2128980">1</span> <span m="2129450">minus</span> <span m="2129930">a2</span> <span m="2131190">alpha to</span> <span m="2131660">the d</span> <span m="2131880">minus</span> <span m="2132550">2</span> <span m="2134420">minus</span> <span m="2134880">ad</span> <span m="2136040">is</span> <span m="2136240">0.</span> <span m="2138600">This</span> <span m="2138870">is</span> <span m="2138990">called</span> <span m="2139460">the</span> <span m="2139720">characteristic</span> <span m="2140630">equation</span> <span m="2141470">of</span> <span m="2141730">the</span> <span m="2141920">recurrence.</span> <span m="2156770">OK?</span> </p>
<p><span m="2158490">Now,</span> <span m="2159490">what</span> <span m="2159840">you</span> <span m="2159920">need</span> <span m="2160090">to</span> <span m="2160150">do</span> <span m="2160570">is</span> <span m="2160690">compute</span> <span m="2161070">the</span> <span m="2161170">roots</span> <span m="2161450">of</span> <span m="2161540">this,</span> <span m="2163590">and</span> <span m="2163750">we'll</span> <span m="2163840">use</span> <span m="2164150">the</span> <span m="2164300">roots</span> <span m="2164710">to</span> <span m="2164800">get</span> <span m="2164990">the</span> <span m="2165070">solution</span> <span m="2165830">to</span> <span m="2166090">the</span> <span m="2166190">recurrence.</span> <span m="2168610">Now,</span> <span m="2168790">the</span> <span m="2168890">simple</span> <span m="2169270">case</span> <span m="2169750">is</span> <span m="2169910">when</span> <span m="2170060">all</span> <span m="2170310">the</span> <span m="2170430">roots</span> <span m="2170780">are</span> <span m="2170920">different--</span> <span m="2172530">so</span> <span m="2172690">let's</span> <span m="2172910">do</span> <span m="2173020">that</span> <span m="2173290">first--</span> <span m="2174840">like</span> <span m="2175020">there</span> <span m="2175120">was in</span> <span m="2175420">the</span> <span m="2175530">Fibonacci</span> <span m="2176060">example.</span> <span m="2181440">All</span> <span m="2181790">d</span> <span m="2182000">roots</span> <span m="2184860">are</span> <span m="2185020">different,</span> <span m="2189470">and</span> <span m="2189690">let's</span> <span m="2189900">call</span> <span m="2190280">them</span> <span m="2191370">alpha</span> <span m="2191800">1</span> <span m="2192300">to</span> <span m="2192800">alpha</span> <span m="2193130">d.</span> </p>
<p><span m="2199610">In</span> <span m="2199730">that</span> <span m="2199950">case,</span> <span m="2200540">the</span> <span m="2201140">solution</span> <span m="2201980">is</span> <span m="2202970">all</span> <span m="2203290">linear</span> <span m="2203600">combinations</span> <span m="2204070">of</span> <span m="2204540">their</span> <span m="2204880">n-th</span> <span m="2205120">powers,</span> <span m="2208850">just</span> <span m="2209220">like</span> <span m="2209400">it</span> <span m="2209490">was</span> <span m="2209780">with</span> <span m="2209850">Fibonacci</span> <span m="2213920">without</span> <span m="2214230">the</span> <span m="2214310">boundary</span> <span m="2214630">conditions.</span> <span m="2216180">c1</span> <span m="2217100">alpha</span> <span m="2217500">1 to the</span> <span m="2217890">n</span> <span m="2218290">plus</span> <span m="2218670">c2</span> <span m="2219870">alpha 2 to the</span> <span m="2220210">n</span> <span m="2222740">plus</span> <span m="2223150">cd</span> <span m="2224590">alpha</span> <span m="2226102">d to the</span> <span m="2226520">n.</span> <span m="2228900">All right?</span> <span m="2229080">Same</span> <span m="2229370">thing</span> <span m="2229650">happens</span> <span m="2230060">that</span> <span m="2230150">happened</span> <span m="2230430">in</span> <span m="2230510">Fibonacci,</span> <span m="2231340">that</span> <span m="2231830">this</span> <span m="2232220">becomes</span> <span m="2232670">the</span> <span m="2232950">solution</span> <span m="2233610">before</span> <span m="2234270">the</span> <span m="2234390">boundary</span> <span m="2234780">conditions</span> <span m="2235250">are</span> <span m="2235350">applied.</span> <span m="2236702">All right?</span> </p>
<p><span m="2250500">And</span> <span m="2250710">by</span> <span m="2250850">the</span> <span m="2250960">way,</span> <span m="2251160">if</span> <span m="2251290">your</span> <span m="2252100">characteristic</span> <span m="2252680">equation</span> <span m="2253220">has</span> <span m="2253730">imaginary</span> <span m="2254550">roots,</span> <span m="2255140">that's</span> <span m="2255400">fine.</span> <span m="2256930">It</span> <span m="2256960">doesn't</span> <span m="2257250">matter</span> <span m="2257590">because</span> <span m="2258260">the</span> <span m="2258690">imaginary</span> <span m="2259270">i</span> <span m="2259480">term</span> <span m="2259780">will</span> <span m="2259880">disappear</span> <span m="2260490">just</span> <span m="2260890">like</span> <span m="2261210">the</span> <span m="2261300">square</span> <span m="2261470">root</span> <span m="2261570">of</span> <span m="2261650">5</span> <span m="2261900">disappeared.</span> <span m="2263350">Has to</span> <span m="2263780">because</span> <span m="2264070">we</span> <span m="2264150">know</span> <span m="2264360">f</span> <span m="2264510">of</span> <span m="2264640">n,</span> <span m="2265740">in</span> <span m="2265870">this</span> <span m="2266040">case,</span> <span m="2266390">is</span> <span m="2266600">a</span> <span m="2266680">real</span> <span m="2266860">number.</span> <span m="2269130">All right?</span> <span m="2271300">OK.</span> <span m="2271610">Now,</span> <span m="2271770">to</span> <span m="2271860">find</span> <span m="2272240">out</span> <span m="2272420">the</span> <span m="2272500">values</span> <span m="2272970">of</span> <span m="2273060">the</span> <span m="2273180">coefficients,</span> <span m="2274450">what</span> <span m="2274600">are</span> <span m="2274650">we</span> <span m="2274740">going</span> <span m="2274830">to</span> <span m="2274900">do?</span> <span m="2277710">What</span> <span m="2278130">is</span> <span m="2278250">it?</span> </p>
<p><span m="2278710">AUDIENCE: Boundary</span> <span m="2279065">conditions.</span> </p>
<p><span m="2279420">PROFESSOR: Boundary</span> <span m="2279820">conditions,</span> <span m="2280360">yep.</span> <span m="2281490">So</span> <span m="2281650">now</span> <span m="2281930">we</span> <span m="2282050">solve</span> <span m="2284850">for</span> <span m="2285530">c1,</span> <span m="2286550">c2,</span> <span m="2288180">cd</span> <span m="2290490">from</span> <span m="2292220">the</span> <span m="2292320">boundary</span> <span m="2292730">conditions.</span> <span m="2293870">That</span> <span m="2294070">is,</span> <span m="2294290">f</span> <span m="2294500">of  i</span> <span m="2295490">is</span> <span m="2296050">b</span> <span m="2296520">of</span> <span m="2296810">i</span> <span m="2298286">for</span> <span m="2298778">i</span> <span m="2299270">from</span> <span m="2299460">0</span> <span m="2299790">to</span> <span m="2299940">d.</span> <span m="2303030">So</span> <span m="2303320">for</span> <span m="2303470">example,</span> <span m="2306220">we</span> <span m="2306250">know</span> <span m="2306490">that</span> <span m="2306660">f</span> <span m="2306840">of</span> <span m="2306930">0</span> <span m="2308960">is</span> <span m="2310350">c1</span> <span m="2311500">times</span> <span m="2312060">alpha</span> <span m="2312380">1</span> <span m="2312580">to the</span> <span m="2312730">0,</span> <span m="2313380">which is</span> <span m="2313740">1,</span> <span m="2314800">c2</span> <span m="2315420">alpha 2</span> <span m="2315650">to</span> <span m="2315840">the</span> <span m="2315940">0</span> <span m="2317960">plus</span> <span m="2318460">cd.</span> <span m="2320480">And</span> <span m="2320630">that's</span> <span m="2320870">going</span> <span m="2320960">to</span> <span m="2321010">be</span> <span m="2321180">equal</span> <span m="2321410">to</span> <span m="2321470">b0</span> <span m="2323670">and</span> <span m="2324080">so</span> <span m="2324290">forth.</span> </p>
<p><span m="2325950">So</span> <span m="2325990">you</span> <span m="2326060">get</span> <span m="2326220">a</span> <span m="2326490">system</span> <span m="2326970">of</span> <span m="2327060">equations,</span> <span m="2327870">d</span> <span m="2328070">equations</span> <span m="2328600">in</span> <span m="2328730">d</span> <span m="2328890">variables,</span> <span m="2329470">that</span> <span m="2329600">you</span> <span m="2329750">solve</span> <span m="2330550">to</span> <span m="2330860">find</span> <span m="2331240">the</span> <span m="2331790">coefficients.</span> <span m="2333820">And</span> <span m="2334010">that</span> <span m="2334180">will</span> <span m="2334310">give</span> <span m="2334950">the</span> <span m="2335150">unique</span> <span m="2336070">and</span> <span m="2336200">correct</span> <span m="2336690">solution</span> <span m="2337170">to the</span> <span m="2337635">recurrence.</span> <span m="2340430">All</span> <span m="2340520">right?</span> <span m="2340860">Now,</span> <span m="2341630">turns</span> <span m="2341930">out</span> <span m="2342290">this</span> <span m="2342490">system</span> <span m="2342750">of</span> <span m="2342830">equations</span> <span m="2343400">is</span> <span m="2343620">never</span> <span m="2344130">degenerate.</span> <span m="2345620">It</span> <span m="2345860">always</span> <span m="2346400">has</span> <span m="2346650">a</span> <span m="2346720">solution.</span> <span m="2347550">And</span> <span m="2347850">you</span> <span m="2347980">can</span> <span m="2348150">prove</span> <span m="2348460">that,</span> <span m="2348650">but</span> <span m="2349020">we</span> <span m="2349110">won't</span> <span m="2349340">do</span> <span m="2349450">that.</span> <span m="2351860">Any</span> <span m="2352020">questions</span> <span m="2354175">about</span> <span m="2354660">how</span> <span m="2354810">to</span> <span m="2354910">do</span> <span m="2355060">that?</span> <span m="2357760">All</span> <span m="2357850">right.</span> <span m="2358420">Yeah?</span> </p>
<p><span m="2359209">AUDIENCE: Sorry,</span> <span m="2359708">I just had a quick</span> <span m="2360207">question.</span> <span m="2360706">But</span> <span m="2361205">what</span> <span m="2361704">exactly</span> <span m="2362203">was the</span> <span m="2362702">distance</span> <span m="2363201">there</span> <span m="2363700">between n and</span> <span m="2364199">d?</span> <span m="2365700">d is</span> <span m="2366000">like</span> <span m="2366210">the</span> <span m="2366710">degree</span> <span m="2367210">of--</span> <span m="2367710">which the</span> <span m="2368210">terms</span> <span m="2368710">you go back</span> <span m="2369210">to.</span> </p>
<p><span m="2369710">PROFESSOR: Yes,</span> <span m="2371210">Yeah.</span> <span m="2371660">d</span> <span m="2372110">is</span> <span m="2372650">how</span> <span m="2372880">far</span> <span m="2373130">back</span> <span m="2373400">you're</span> <span m="2373490">going.</span> <span m="2375260">You</span> <span m="2375480">go</span> <span m="2375700">as</span> <span m="2375920">far</span> <span m="2376230">back</span> <span m="2376710">as</span> <span m="2377600">f</span> <span m="2377860">of</span> <span m="2378040">n</span> <span m="2378310">minus</span> <span m="2378730">d.</span> </p>
<p><span m="2379420">AUDIENCE: OK.</span> <span m="2379880">Cool.</span> </p>
<p><span m="2380340">PROFESSOR: And</span> <span m="2380470">that</span> <span m="2380640">becomes</span> <span m="2380930">the</span> <span m="2381000">degree,</span> <span m="2381470">and that</span> <span m="2381730">is</span> <span m="2381920">a</span> <span m="2382020">constant--</span> <span m="2382890">2,</span> <span m="2383250">3,</span> <span m="2383770">4.</span> <span m="2385310">Well, I</span> <span m="2385450">don't think</span> <span m="2385590">we'll</span> <span m="2385740">ever</span> <span m="2385940">ask</span> <span m="2386190">you</span> <span m="2386620">anything</span> <span m="2386970">beyond</span> <span m="2387250">4</span> <span m="2387510">because</span> <span m="2387680">it</span> <span m="2387750">gets</span> <span m="2387890">to</span> <span m="2387960">be</span> <span m="2388070">a</span> <span m="2388120">pain</span> <span m="2389350">to</span> <span m="2389470">do.</span> <span m="2389660">Typically, it's</span> <span m="2390100">2</span> <span m="2390350">or</span> <span m="2390430">3,</span> <span m="2391850">sometimes</span> <span m="2392300">even</span> <span m="2392500">just</span> <span m="2392720">1.</span> <span m="2393450">And</span> <span m="2393620">then</span> <span m="2393730">that</span> <span m="2393890">becomes</span> <span m="2394250">the</span> <span m="2394330">power</span> <span m="2395070">of</span> <span m="2395450">your</span> <span m="2395730">characteristic</span> <span m="2396290">equation.</span> <span m="2398010">It's</span> <span m="2398100">the</span> <span m="2398230">order</span> <span m="2398590">of</span> <span m="2398650">the</span> <span m="2398740">characteristic</span> <span m="2399180">equation.</span> <span m="2401940">Any</span> <span m="2402130">other</span> <span m="2402310">questions?</span> </p>
<p><span m="2405590">All right.</span> <span m="2405740">That</span> <span m="2405960">was</span> <span m="2406120">the</span> <span m="2406170">nice</span> <span m="2406700">case</span> <span m="2407120">where</span> <span m="2407500">all</span> <span m="2407790">the</span> <span m="2407920">roots</span> <span m="2408550">of</span> <span m="2408700">your</span> <span m="2408840">characteristic</span> <span m="2409820">equation</span> <span m="2410450">were</span> <span m="2410460">different.</span> <span m="2411660">If</span> <span m="2411960">they're</span> <span m="2412090">not</span> <span m="2412360">all</span> <span m="2412530">different,</span> <span m="2413670">it's</span> <span m="2413870">a</span> <span m="2413920">little</span> <span m="2414100">more</span> <span m="2414250">complicated.</span> <span m="2433380">So</span> <span m="2433620">the</span> <span m="2433730">tricky</span> <span m="2434060">case</span> <span m="2440370">is</span> <span m="2441170">repeated</span> <span m="2441670">roots.</span> </p>
<p><span m="2449320">Now,</span> <span m="2450850">the</span> <span m="2450970">theorem,</span> <span m="2451550">which</span> <span m="2451750">we</span> <span m="2451840">won't</span> <span m="2452090">prove</span> <span m="2452560">but</span> <span m="2452660">tells</span> <span m="2452900">you</span> <span m="2453090">what</span> <span m="2453260">to</span> <span m="2453360">do,</span> <span m="2454550">is</span> <span m="2454860">that</span> <span m="2455990">if</span> <span m="2456260">alpha</span> <span m="2457590">is</span> <span m="2457860">a</span> <span m="2458357">root</span> <span m="2460842">of</span> <span m="2461340">the</span> <span m="2461500">characteristic</span> <span m="2462290">equation</span> <span m="2470460">and</span> <span m="2470670">it is</span> <span m="2470890">repeated</span> <span m="2471520">r</span> <span m="2471820">times--</span> <span m="2472660">so</span> <span m="2473850">x</span> <span m="2474190">minus</span> <span m="2474540">alpha</span> <span m="2474850">to</span> <span m="2474900">the</span> <span m="2475020">r-th</span> <span m="2475270">power</span> <span m="2476530">is</span> <span m="2476690">a</span> <span m="2476740">factor--</span> <span m="2492450">then</span> <span m="2495220">alpha to the</span> <span m="2495650">n,</span> <span m="2496958">n</span> <span m="2497394">times</span> <span m="2497830">alpha to the</span> <span m="2498130">n,</span> <span m="2499576">n</span> <span m="2500060">squared</span> <span m="2500900">times alpha</span> <span m="2501195">to the</span> <span m="2501490">n,</span> <span m="2502436">all</span> <span m="2502910">the</span> <span m="2503010">way</span> <span m="2503190">up</span> <span m="2503350">to</span> <span m="2504420">n</span> <span m="2504720">to</span> <span m="2504810">the</span> <span m="2504990">r</span> <span m="2505300">minus</span> <span m="2505700">1</span> <span m="2506070">times</span> <span m="2506370">alpha to the</span> <span m="2506650">n</span> <span m="2507790">are</span> <span m="2508050">all</span> <span m="2508420">solutions</span> <span m="2509110">to</span> <span m="2509210">the</span> <span m="2509310">recurrence.</span> </p>
<p><span m="2524610">All</span> <span m="2524760">right?</span> <span m="2524960">And</span> <span m="2525080">then</span> <span m="2525410">you</span> <span m="2525550">would</span> <span m="2526040">treat</span> <span m="2526450">them</span> <span m="2526890">just</span> <span m="2527090">as</span> <span m="2527210">you</span> <span m="2527370">would</span> <span m="2527630">the</span> <span m="2527790">other</span> <span m="2527970">roots.</span> <span m="2528420">You</span> <span m="2528530">take</span> <span m="2528920">linear</span> <span m="2529230">combinations,</span> <span m="2530140">just</span> <span m="2530380">like</span> <span m="2530540">you</span> <span m="2530630">did</span> <span m="2530830">with</span> <span m="2530940">the</span> <span m="2531050">other</span> <span m="2531230">roots</span> <span m="2532760">to</span> <span m="2532840">put</span> <span m="2533000">it</span> <span m="2533080">all</span> <span m="2533260">together.</span> <span m="2537470">By</span> <span m="2537590">the</span> <span m="2537690">way,</span> <span m="2538040">is</span> <span m="2538190">anybody</span> <span m="2538520">starting</span> <span m="2538770">to</span> <span m="2538910">recognize</span> <span m="2541110">a</span> <span m="2541410">similarity</span> <span m="2542020">with</span> <span m="2542160">something</span> <span m="2542470">else</span> <span m="2542740">that</span> <span m="2542840">you've</span> <span m="2543340">studied</span> <span m="2543740">in</span> <span m="2543810">the</span> <span m="2543890">past?</span> </p>
<p><span m="2544782">AUDIENCE: Differential</span> <span m="2545184">equations.</span> </p>
<p><span m="2545990">PROFESSOR: Differential</span> <span m="2546490">equations.</span> <span m="2547910">This</span> <span m="2548170">is</span> <span m="2548450">the</span> <span m="2548700">discrete</span> <span m="2549130">analog</span> <span m="2550170">of</span> <span m="2550440">differential</span> <span m="2550960">equations.</span> <span m="2552576">All right?</span> <span m="2553050">Recurrences</span> <span m="2554060">is</span> <span m="2554280">the</span> <span m="2554390">same</span> <span m="2555010">thing.</span> <span m="2555390">All</span> <span m="2555620">the</span> <span m="2555700">math</span> <span m="2555970">we're</span> <span m="2556060">going</span> <span m="2556140">to</span> <span m="2556190">do</span> <span m="2556350">henceforth</span> <span m="2557450">is</span> <span m="2557650">going</span> <span m="2557740">to</span> <span m="2557780">look</span> <span m="2558000">just</span> <span m="2558480">like</span> <span m="2558630">what</span> <span m="2558790">you</span> <span m="2558910">did</span> <span m="2559840">with</span> <span m="2560670">differential</span> <span m="2561150">equations.</span> <span m="2562510">So</span> <span m="2562520">that's</span> <span m="2563020">sort</span> <span m="2563230">of</span> <span m="2563290">good</span> <span m="2563480">news</span> <span m="2563680">and,</span> <span m="2563880">I</span> <span m="2563940">guess,</span> <span m="2564160">bad</span> <span m="2564420">news,</span> <span m="2564690">depending</span> <span m="2564950">on</span> <span m="2565070">whether</span> <span m="2565250">you</span> <span m="2565350">like</span> <span m="2565570">that stuff.</span> </p>
<p><span m="2565820">AUDIENCE: Bad news.</span> </p>
<p><span m="2566820">PROFESSOR: Bad</span> <span m="2567090">news.</span> <span m="2567430">Yeah,</span> <span m="2567810">OK.</span> <span m="2571810">All right,</span> <span m="2572000">let's</span> <span m="2572200">do</span> <span m="2572310">an</span> <span m="2572380">example</span> <span m="2573560">that</span> <span m="2573870">uses</span> <span m="2574270">maybe</span> <span m="2574520">the</span> <span m="2574610">repeated</span> <span m="2575240">roots</span> <span m="2575510">case.</span> <span m="2577480">Suppose</span> <span m="2577910">there's</span> <span m="2578100">a</span> <span m="2578180">plant</span> <span m="2578840">out</span> <span m="2579100">there,</span> <span m="2580140">and</span> <span m="2580420">this</span> <span m="2580600">plant</span> <span m="2581300">lives</span> <span m="2581670">forever</span> <span m="2583150">but</span> <span m="2583330">only</span> <span m="2583720">reproduces</span> <span m="2584510">in</span> <span m="2584630">the</span> <span m="2584720">first</span> <span m="2585150">year</span> <span m="2585360">of</span> <span m="2585450">life</span> <span m="2586270">and</span> <span m="2586430">then</span> <span m="2586610">never</span> <span m="2586940">again.</span> <span m="2588070">And</span> <span m="2588250">it</span> <span m="2588340">reproduces</span> <span m="2589040">one</span> <span m="2589430">for</span> <span m="2589560">one.</span> <span m="2591030">All right?</span> <span m="2591500">Let's</span> <span m="2591770">see</span> <span m="2592370">what</span> <span m="2592520">happens</span> <span m="2592870">in</span> <span m="2592950">that</span> <span m="2593150">case.</span> </p>
<p><span m="2599890">Actually,</span> <span m="2600180">there's</span> <span m="2600310">a</span> <span m="2600390">plant</span> <span m="2600840">sort</span> <span m="2601090">of</span> <span m="2601180">like</span> <span m="2601400">this</span> <span m="2601620">in</span> <span m="2601730">Hawaii.</span> <span m="2602310">I</span> <span m="2602380">think</span> <span m="2602590">it's</span> <span m="2602720">called</span> <span m="2602930">the</span> <span m="2603030">century</span> <span m="2603540">plant,</span> <span m="2605460">and</span> <span m="2605660">it's</span> <span m="2606100">rare.</span> <span m="2607570">We'll</span> <span m="2607710">see</span> <span m="2607890">why</span> <span m="2608230">when</span> <span m="2608390">we</span> <span m="2608470">solve</span> <span m="2608780">this</span> <span m="2608920">problem.</span> <span m="2611990">So</span> <span m="2612160">it</span> <span m="2612250">reproduces</span> <span m="2612920">one</span> <span m="2613240">for</span> <span m="2613380">one</span> <span m="2615060">during</span> <span m="2617410">the</span> <span m="2617570">first</span> <span m="2617970">year</span> <span m="2618180">of</span> <span m="2618240">life,</span> <span m="2626290">then</span> <span m="2626440">never</span> <span m="2626840">again,</span> <span m="2635120">and</span> <span m="2635720">the</span> <span m="2635810">plant</span> <span m="2636260">lives</span> <span m="2636540">forever.</span> <span m="2645500">So</span> <span m="2645680">our</span> <span m="2645840">question</span> <span m="2646370">is,</span> <span m="2647190">how</span> <span m="2647470">fast</span> <span m="2647970">does</span> <span m="2648120">the</span> <span m="2648220">plant</span> <span m="2648730">population</span> <span m="2650170">grow?</span> <span m="2651065">All right?</span> <span m="2651450">In</span> <span m="2651750">year</span> <span m="2652040">n,</span> <span m="2652230">how</span> <span m="2652360">many</span> <span m="2652560">plants</span> <span m="2652920">there</span> <span m="2653110">are.</span> </p>
<p><span m="2655010">So</span> <span m="2655310">to</span> <span m="2655600">figure</span> <span m="2655800">that</span> <span m="2656040">out,</span> <span m="2656785">we're</span> <span m="2657220">going to</span> <span m="2657300">set</span> <span m="2657490">up</span> <span m="2657620">a</span> <span m="2657740">recurrence.</span> <span m="2661200">We're</span> <span m="2661320">going</span> <span m="2661410">to</span> <span m="2661500">let</span> <span m="2661950">f</span> <span m="2662160">of</span> <span m="2662290">n</span> <span m="2664576">be</span> <span m="2665040">the</span> <span m="2665140">number</span> <span m="2665390">of</span> <span m="2665470">plants</span> <span m="2665950">in year</span> <span m="2666030">n.</span> <span m="2673150">And</span> <span m="2673540">we're</span> <span m="2673640">going</span> <span m="2673740">to</span> <span m="2673820">say</span> <span m="2674110">that</span> <span m="2674240">the</span> <span m="2674330">first</span> <span m="2674820">plant</span> <span m="2675670">miraculously</span> <span m="2676380">comes</span> <span m="2676610">into</span> <span m="2676810">existence</span> <span m="2677330">in</span> <span m="2677410">year</span> <span m="2677580">1.</span> <span m="2679090">So</span> <span m="2679230">that</span> <span m="2679370">in</span> <span m="2679480">year</span> <span m="2679720">0</span> <span m="2680200">there's</span> <span m="2680440">none</span> <span m="2680670">of</span> <span m="2680780">them,</span> <span m="2682110">and</span> <span m="2682390">in</span> <span m="2682470">year</span> <span m="2682730">1</span> <span m="2683060">there's</span> <span m="2683250">1.</span> <span m="2685730">And</span> <span m="2685870">now</span> <span m="2686050">we</span> <span m="2686120">want</span> <span m="2686330">to</span> <span m="2686380">know</span> <span m="2686580">how</span> <span m="2686800">many</span> <span m="2686980">there</span> <span m="2687210">are</span> <span m="2687360">in</span> <span m="2687420">year n.</span> <span m="2705540">OK.</span> <span m="2706070">So</span> <span m="2706310">let's</span> <span m="2706580">set</span> <span m="2706730">up</span> <span m="2706860">the</span> <span m="2706950">recurrence.</span> <span m="2710190">f</span> <span m="2710400">of</span> <span m="2710540">n</span> <span m="2711160">equals--</span> <span m="2711840">well,</span> <span m="2713070">the</span> <span m="2713180">previous</span> <span m="2713820">year,</span> <span m="2715200">how</span> <span m="2715380">many</span> <span m="2715580">plants</span> <span m="2715930">were</span> <span m="2716100">there</span> <span m="2718240">in</span> <span m="2718280">terms</span> <span m="2718490">of</span> <span m="2718580">f?</span> </p>
<p><span m="2720540">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="2721030">PROFESSOR: f</span> <span m="2721520">of</span> <span m="2721630">n</span> <span m="2721760">minus</span> <span m="2722090">1.</span> <span m="2723500">All right?</span> <span m="2723800">That's</span> <span m="2724010">how</span> <span m="2724090">many</span> <span m="2724270">there</span> <span m="2724380">were</span> <span m="2724560">last</span> <span m="2724910">year,</span> <span m="2725060">and</span> <span m="2725120">they're</span> <span m="2725270">all</span> <span m="2725440">still</span> <span m="2725660">alive,</span> <span m="2726650">plus</span> <span m="2727120">we've</span> <span m="2727250">got</span> <span m="2727430">to</span> <span m="2727660">add</span> <span m="2727960">the</span> <span m="2728020">number</span> <span m="2728290">of</span> <span m="2728390">new</span> <span m="2728670">plants.</span> <span m="2731360">Well,</span> <span m="2731900">that</span> <span m="2732160">would</span> <span m="2732310">be--</span> <span m="2734400">yeah,</span> <span m="2734720">well,</span> <span m="2734810">how</span> <span m="2734930">many</span> <span m="2735110">new</span> <span m="2735360">plants</span> <span m="2735820">are</span> <span m="2735980">there?</span> <span m="2736330">It's</span> <span m="2736520">all</span> <span m="2736750">the</span> <span m="2736830">plants</span> <span m="2737300">that</span> <span m="2737430">were</span> <span m="2737680">one-year-old</span> <span m="2738440">last</span> <span m="2738900">year.</span> </p>
<p><span m="2740560">So</span> <span m="2743280">that's</span> <span m="2743480">all</span> <span m="2743670">the</span> <span m="2743760">ones</span> <span m="2744010">that were</span> <span m="2744130">alive</span> <span m="2744650">last</span> <span m="2745020">year</span> <span m="2745230">minus</span> <span m="2745800">the</span> <span m="2745910">ones</span> <span m="2746240">that were</span> <span m="2746340">alive</span> <span m="2746830">the</span> <span m="2746930">year</span> <span m="2747100">before.</span> <span m="2750450">That's</span> <span m="2750700">how</span> <span m="2750810">many</span> <span m="2751010">new</span> <span m="2751230">plants</span> <span m="2751630">there</span> <span m="2751750">were</span> <span m="2752100">last</span> <span m="2752460">year,</span> <span m="2752850">and</span> <span m="2752960">they</span> <span m="2753100">each</span> <span m="2753290">produce</span> <span m="2753600">1</span> <span m="2754130">new</span> <span m="2754280">one</span> <span m="2754520">this year.</span> <span m="2756186">Is that</span> <span m="2756650">OK?</span> <span m="2756915">Everybody</span> <span m="2757180">buy</span> <span m="2757310">that?</span> <span m="2760510">OK.</span> <span m="2763500">So</span> <span m="2763830">that</span> <span m="2764080">equals</span> <span m="2764760">2</span> <span m="2765450">f</span> <span m="2765650">of</span> <span m="2765780">n</span> <span m="2765940">minus</span> <span m="2766280">1</span> <span m="2767770">minus</span> <span m="2768960">f</span> <span m="2769180">of</span> <span m="2769280">n</span> <span m="2769420">minus</span> <span m="2769720">2.</span> <span m="2774840">OK.</span> <span m="2776280">What's</span> <span m="2776670">my</span> <span m="2776820">characteristic</span> <span m="2777540">equation</span> <span m="2777845">for</span> <span m="2778150">this</span> <span m="2778649">recurrence?</span> <span m="2784140">Yeah?</span> </p>
<p><span m="2784680">AUDIENCE: Alpha</span> <span m="2785156">squared</span> <span m="2787060">minus</span> <span m="2787536">2 alpha</span> <span m="2788012">plus</span> <span m="2788488">1.</span> </p>
<p><span m="2789440">PROFESSOR: Yes.</span> <span m="2791344">OK?</span> <span m="2792180">alpha</span> <span m="2792530">squared</span> <span m="2794640">minus</span> <span m="2795270">2 alpha</span> <span m="2798020">plus</span> <span m="2798610">1.</span> <span m="2800260">All right?</span> <span m="2800430">We</span> <span m="2800570">jumped</span> <span m="2800980">to</span> <span m="2801150">that</span> <span m="2801340">answer.</span> <span m="2801730">I</span> <span m="2801780">could</span> <span m="2801980">have</span> <span m="2802120">written</span> <span m="2802380">down</span> <span m="2802980">alpha to the</span> <span m="2803370">n</span> <span m="2804490">minus</span> <span m="2805040">2</span> <span m="2805410">times</span> <span m="2805780">alpha to the</span> <span m="2806030">n</span> <span m="2806320">minus</span> <span m="2806650">1</span> <span m="2807590">plus</span> <span m="2808080">alpha to the</span> <span m="2808440">n</span> <span m="2808590">minus</span> <span m="2808890">2</span> <span m="2809180">and</span> <span m="2809250">then</span> <span m="2809370">divided</span> <span m="2809850">by alpha to the</span> <span m="2809990">n</span> <span m="2810380">minus</span> <span m="2810660">2.</span> <span m="2810930">But</span> <span m="2811420">pretty</span> <span m="2811630">quickly, you</span> <span m="2811990">want</span> <span m="2812160">to</span> <span m="2812220">start</span> <span m="2812500">just</span> <span m="2813270">reading</span> <span m="2813530">it</span> <span m="2813600">off.</span> </p>
<p><span m="2814570">That's</span> <span m="2814750">the</span> <span m="2814830">characteristic</span> <span m="2815280">equation.</span> <span m="2816420">I</span> <span m="2816520">get</span> <span m="2816820">alpha</span> <span m="2817140">squared,</span> <span m="2818160">and then</span> <span m="2818290">bringing</span> <span m="2818500">this</span> <span m="2818650">over, a</span> <span m="2818870">minus</span> <span m="2819210">2</span> <span m="2819430">alpha,</span> <span m="2820080">bring</span> <span m="2820290">this</span> <span m="2820630">over,</span> <span m="2820890">plus</span> <span m="2821440">1</span> <span m="2822360">equals</span> <span m="2822700">0.</span> <span m="2825680">Any</span> <span m="2825780">questions</span> <span m="2827150">on</span> <span m="2827300">that?</span> <span m="2830000">This</span> <span m="2830210">is</span> <span m="2830390">the</span> <span m="2830470">process</span> <span m="2830990">now</span> <span m="2831180">you</span> <span m="2831360">will</span> <span m="2831500">always</span> <span m="2831790">use</span> <span m="2832090">that</span> <span m="2832160">we're</span> <span m="2832250">going</span> <span m="2832490">through.</span> <span m="2833780">All right.</span> <span m="2833920">What</span> <span m="2834140">are</span> <span m="2834170">the</span> <span m="2834310">roots to this</span> <span m="2834610">thing?</span> <span m="2838410">What</span> <span m="2838570">are</span> <span m="2838590">the</span> <span m="2838750">roots to</span> <span m="2839070">the</span> <span m="2839180">characteristic</span> <span m="2839670">equation?</span> </p>
<p><span m="2840200">AUDIENCE: Alpha</span> <span m="2840666">minus</span> <span m="2841132">1</span> <span m="2841598">[INAUDIBLE].</span> </p>
<p><span m="2844400">PROFESSOR: Yeah.</span> <span m="2844850">Alpha</span> <span m="2845150">minus</span> <span m="2845490">1</span> <span m="2846030">times</span> <span m="2846250">alpha</span> <span m="2846450">minus</span> <span m="2846690">1</span> <span m="2846850">is</span> <span m="2846990">0,</span> <span m="2847470">which</span> <span m="2847670">means</span> <span m="2847990">alpha</span> <span m="2848480">equals?</span> </p>
<p><span m="2849380">AUDIENCE: 1.</span> </p>
<p><span m="2849800">PROFESSOR: 1.</span> <span m="2850830">So</span> <span m="2851050">the</span> <span m="2851180">roots</span> <span m="2851510">are</span> <span m="2852300">alpha</span> <span m="2852750">equals</span> <span m="2853160">1,</span> <span m="2854020">and</span> <span m="2854390">it's</span> <span m="2854540">a</span> <span m="2854640">double</span> <span m="2855110">root.</span> <span m="2857590">OK?</span> <span m="2859070">It's</span> <span m="2859230">a</span> <span m="2859290">double</span> <span m="2859670">root.</span> <span m="2860040">It</span> <span m="2860270">occurs</span> <span m="2860630">twice</span> <span m="2863320">because</span> <span m="2863700">this</span> <span m="2863990">is</span> <span m="2864290">alpha</span> <span m="2864560">minus</span> <span m="2864860">1</span> <span m="2865440">squared.</span> <span m="2868110">OK.</span> <span m="2869550">So</span> <span m="2869840">what</span> <span m="2870390">solutions</span> <span m="2871340">am</span> <span m="2871440">I</span> <span m="2871530">going</span> <span m="2871680">to</span> <span m="2871720">use</span> <span m="2873701">to</span> <span m="2874090">my</span> <span m="2874230">recurrence?</span> <span m="2880330">What's</span> <span m="2880690">f</span> <span m="2880910">of</span> <span m="2881020">n</span> <span m="2881270">going</span> <span m="2881380">to</span> <span m="2881450">be</span> <span m="2882630">now</span> <span m="2883830">before</span> <span m="2884290">I</span> <span m="2884350">put</span> <span m="2884570">in</span> <span m="2884730">the</span> <span m="2884870">boundary</span> <span m="2885210">conditions?</span> <span m="2887466">Well,</span> <span m="2887870">it's</span> <span m="2887970">going</span> <span m="2888050">to</span> <span m="2888090">be</span> <span m="2888190">c1</span> <span m="2888890">times</span> <span m="2890220">what</span> <span m="2890480">to</span> <span m="2890570">the</span> <span m="2890720">n?</span> </p>
<p><span m="2893516">AUDIENCE: 1.</span> </p>
<p><span m="2894448">PROFESSOR: 1.</span> <span m="2895140">That's</span> <span m="2895370">the</span> <span m="2895450">first</span> <span m="2895750">root.</span> <span m="2896960">What's</span> <span m="2897290">here?</span> <span m="2897600">What's</span> <span m="2897830">the</span> <span m="2897950">next</span> <span m="2898330">thing</span> <span m="2898500">I'm</span> <span m="2898570">going</span> <span m="2898670">to</span> <span m="2898740">put</span> <span m="2898950">here?</span> </p>
<p><span m="2901780">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2902740">PROFESSOR: n</span> <span m="2903150">times</span> <span m="2903410">1</span> <span m="2903690">to</span> <span m="2903770">the</span> <span m="2903940">n.</span> <span m="2904555">All</span> <span m="2904810">right?</span> <span m="2907540">Because</span> <span m="2907980">I've</span> <span m="2908100">got</span> <span m="2908260">a</span> <span m="2908370">root</span> <span m="2908630">that's</span> <span m="2908800">repeated</span> <span m="2909580">twice,</span> <span m="2910150">r</span> <span m="2910420">equals</span> <span m="2910710">2,</span> <span m="2911770">so</span> <span m="2911920">I</span> <span m="2912010">have</span> <span m="2912280">alpha</span> <span m="2912580">to</span> <span m="2912640">the</span> <span m="2912780">n</span> <span m="2913190">and</span> <span m="2913390">n</span> <span m="2913620">alpha to the</span> <span m="2913960">n.</span> <span m="2914790">Just</span> <span m="2915020">happens</span> <span m="2915300">alpha's</span> <span m="2915630">1,</span> <span m="2916120">which</span> <span m="2916350">makes</span> <span m="2916590">it</span> <span m="2916810">really</span> <span m="2917120">easy.</span> <span m="2920080">All</span> <span m="2920180">right?</span> <span m="2920270">So</span> <span m="2920350">this</span> <span m="2920630">is</span> <span m="2920760">now</span> <span m="2920990">my</span> <span m="2921160">solution</span> <span m="2921760">before</span> <span m="2922150">the</span> <span m="2922250">boundary</span> <span m="2922690">conditions.</span> <span m="2926270">In fact,</span> <span m="2926420">that</span> <span m="2926620">gets</span> <span m="2926840">even</span> <span m="2927070">simpler.</span> <span m="2928720">That's</span> <span m="2928920">1.</span> <span m="2929130">That's</span> <span m="2929370">c1</span> <span m="2929870">plus</span> <span m="2930170">c2</span> <span m="2930850">times n.</span> </p>
<p><span m="2934520">And</span> <span m="2934660">now</span> <span m="2934860">all</span> <span m="2935180">that's</span> <span m="2935390">left</span> <span m="2935780">is</span> <span m="2935930">to</span> <span m="2936000">plug in</span> <span m="2936420">the</span> <span m="2936500">boundary</span> <span m="2936890">conditions.</span> <span m="2950680">So</span> <span m="2950850">let's</span> <span m="2951090">do</span> <span m="2951240">that.</span> <span m="2954140">Well,</span> <span m="2954440">f</span> <span m="2954650">of</span> <span m="2954740">0</span> <span m="2956500">equals</span> <span m="2956900">0.</span> <span m="2961330">If</span> <span m="2961430">I</span> <span m="2961510">plug</span> <span m="2961780">it</span> <span m="2961840">into</span> <span m="2961970">here,</span> <span m="2962390">what</span> <span m="2962530">happens</span> <span m="2962910">for</span> <span m="2963050">f</span> <span m="2963160">of</span> <span m="2963260">0?</span> <span m="2963530">What</span> <span m="2963650">does</span> <span m="2963770">that</span> <span m="2963940">become?</span> </p>
<p><span m="2964430">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="2965800">PROFESSOR: C1.</span> <span m="2967430">So</span> <span m="2967730">c1</span> <span m="2968150">equals</span> <span m="2968410">0.</span> <span m="2968690">That's</span> <span m="2968940">good.</span> <span m="2969990">f</span> <span m="2970230">of</span> <span m="2970340">1</span> <span m="2971270">equals</span> <span m="2971640">1.</span> <span m="2972990">And</span> <span m="2973410">now</span> <span m="2973640">that's</span> <span m="2973890">going</span> <span m="2974000">to</span> <span m="2974070">be</span> <span m="2974380">c1</span> <span m="2975070">plus</span> <span m="2975450">c2.</span> <span m="2977060">c1</span> <span m="2977310">is</span> <span m="2977470">0,</span> <span m="2979160">so</span> <span m="2979420">it</span> <span m="2979460">means</span> <span m="2979660">c2</span> <span m="2979870">equals</span> <span m="2980290">1.</span> <span m="2983080">All right.</span> <span m="2983530">So</span> <span m="2983640">this</span> <span m="2983750">is</span> <span m="2983860">really</span> <span m="2984140">easy.</span> <span m="2985390">What's</span> <span m="2985620">f</span> <span m="2985780">of</span> <span m="2985880">n</span> <span m="2986060">equal?</span> <span m="2988430">n.</span> <span m="2990520">So</span> <span m="2990680">we</span> <span m="2990770">went</span> <span m="2990920">through</span> <span m="2991000">a</span> <span m="2991060">lot</span> <span m="2991300">of</span> <span m="2991420">work</span> <span m="2992560">to</span> <span m="2992670">get</span> <span m="2992840">an</span> <span m="2992940">answer</span> <span m="2993350">that</span> <span m="2993450">probably we could</span> <span m="2993880">have</span> <span m="2994210">guessed</span> <span m="2994550">pretty</span> <span m="2994830">easily</span> <span m="2995230">by</span> <span m="2995340">just</span> <span m="2995940">plugging</span> <span m="2996330">in</span> <span m="2996420">a</span> <span m="2996470">few</span> <span m="2996680">examples.</span> </p>
<p><span m="2998150">But</span> <span m="2998170">the</span> <span m="2998250">nice</span> <span m="2998490">thing</span> <span m="2998670">is</span> <span m="2998890">this</span> <span m="2999080">works</span> <span m="2999400">for</span> <span m="2999560">all</span> <span m="2999710">the</span> <span m="2999790">cases</span> <span m="3000190">when</span> <span m="3000410">it's</span> <span m="3000590">not</span> <span m="3000820">so</span> <span m="3000980">easy</span> <span m="3001260">to</span> <span m="3001330">guess</span> <span m="3002490">by</span> <span m="3002680">plugging</span> <span m="3003060">in</span> <span m="3003140">the</span> <span m="3003250">examples.</span> <span m="3005890">Yeah,</span> <span m="3006290">now</span> <span m="3006400">you</span> <span m="3006510">can</span> <span m="3006660">see</span> <span m="3006790">why</span> <span m="3006940">this</span> <span m="3007150">is</span> <span m="3007220">a</span> <span m="3007310">relatively</span> <span m="3007810">rare</span> <span m="3008230">plant.</span> <span m="3009850">Its</span> <span m="3010500">population</span> <span m="3011020">is</span> <span m="3011110">growing</span> <span m="3011330">very</span> <span m="3011600">slowly.</span> <span m="3014990">OK.</span> <span m="3015540">Any</span> <span m="3015760">questions</span> <span m="3016190">about</span> <span m="3016440">that?</span> <span m="3017190">That</span> <span m="3017290">had</span> <span m="3017520">repeated</span> <span m="3017990">roots.</span> <span m="3021790">All</span> <span m="3021970">right.</span> <span m="3022610">Yeah?</span> </p>
<p><span m="3023020">AUDIENCE: So we</span> <span m="3023520">just</span> <span m="3024020">guessed that</span> <span m="3024360">f of n</span> <span m="3024700">is equal  to</span> <span m="3025198">alpha</span> <span m="3025696">to the</span> <span m="3026194">n,</span> <span m="3026692">and we don't know if</span> <span m="3027190">that works</span> <span m="3027688">for like</span> <span m="3028186">different</span> <span m="3028684">[INAUDIBLE].</span> <span m="3029680">But how do you</span> <span m="3030178">know that it</span> <span m="3030676">works until</span> <span m="3031174">you plug  it</span> <span m="3031672">in?</span> <span m="3032170">If you</span> <span m="3032668">tried it out</span> <span m="3033166">[INAUDIBLE]?</span> </p>
<p><span m="3035170">PROFESSOR: Yeah.</span> <span m="3037390">We</span> <span m="3037510">didn't</span> <span m="3037760">go</span> <span m="3037920">through</span> <span m="3038140">and</span> <span m="3038220">prove it.</span> <span m="3038710">We</span> <span m="3039020">used</span> <span m="3039330">some</span> <span m="3039450">facts</span> <span m="3039820">along</span> <span m="3040100">the</span> <span m="3040190">way</span> <span m="3040380">that</span> <span m="3040480">we</span> <span m="3040570">didn't</span> <span m="3040790">prove,</span> <span m="3042750">which</span> <span m="3042980">you</span> <span m="3043080">can</span> <span m="3043400">do.</span> <span m="3043590">Like,</span> <span m="3043780">in</span> <span m="3043920">particular,</span> <span m="3044420">the</span> <span m="3044520">fact</span> <span m="3044860">we</span> <span m="3044960">didn't</span> <span m="3045000">prove</span> <span m="3045310">that</span> <span m="3045660">theorem</span> <span m="3046350">there</span> <span m="3047100">that</span> <span m="3047210">those</span> <span m="3047410">will</span> <span m="3047470">be</span> <span m="3047560">the</span> <span m="3047670">roots.</span> <span m="3048800">And</span> <span m="3048900">we</span> <span m="3048990">didn't</span> <span m="3049150">prove</span> <span m="3049410">that</span> <span m="3049530">any</span> <span m="3049710">linear</span> <span m="3050050">combination</span> <span m="3050940">would</span> <span m="3051100">be</span> <span m="3051230">a</span> <span m="3051290">solution.</span> <span m="3053100">But</span> <span m="3053300">those</span> <span m="3053520">are</span> <span m="3053750">facts</span> <span m="3054090">you</span> <span m="3054170">could</span> <span m="3054350">take</span> <span m="3054630">from--</span> <span m="3054790">you</span> <span m="3054890">could</span> <span m="3055040">actually</span> <span m="3055280">prove them. They're</span> <span m="3055530">not</span> <span m="3055840">too</span> <span m="3055990">hard</span> <span m="3056190">to</span> <span m="3056250">prove.</span> </p>
<p><span m="3057490">But</span> <span m="3057650">otherwise,</span> <span m="3058590">I</span> <span m="3058670">think</span> <span m="3058870">we</span> <span m="3058990">went</span> <span m="3059180">through</span> <span m="3059290">every</span> <span m="3059470">step</span> <span m="3059760">and</span> <span m="3059910">narrowed</span> <span m="3060320">it</span> <span m="3060410">in.</span> <span m="3061190">If</span> <span m="3061320">we</span> <span m="3061380">wanted</span> <span m="3061540">to</span> <span m="3061580">be</span> <span m="3061700">really</span> <span m="3062100">sure,</span> <span m="3062410">we'd</span> <span m="3062530">go</span> <span m="3062700">back</span> <span m="3062960">and</span> <span m="3063070">prove it</span> <span m="3063330">by</span> <span m="3063500">induction</span> <span m="3063990">against</span> <span m="3064280">the</span> <span m="3064390">original</span> <span m="3064730">recurrence.</span> <span m="3066020">And</span> <span m="3066220">we'd</span> <span m="3066380">say</span> <span m="3067600">this</span> <span m="3067870">would</span> <span m="3067970">be</span> <span m="3068090">the</span> <span m="3068180">induction</span> <span m="3068530">hypothesis.</span> <span m="3070030">We'd</span> <span m="3070230">cover</span> <span m="3070460">the</span> <span m="3070560">base</span> <span m="3070860">cases--</span> <span m="3071220">f</span> <span m="3071380">of</span> <span m="3071460">0</span> <span m="3071730">and</span> <span m="3071810">f</span> <span m="3071940">of</span> <span m="3072030">1.</span> </p>
<p><span m="3072930">And</span> <span m="3073000">then</span> <span m="3073110">we'd</span> <span m="3073250">plug</span> <span m="3073600">this</span> <span m="3073880">into</span> <span m="3074400">the</span> <span m="3074510">recurrence</span> <span m="3075060">up</span> <span m="3075220">there,</span> <span m="3076610">which</span> <span m="3077010">is,</span> <span m="3078180">does</span> <span m="3078570">n</span> <span m="3078940">equal</span> <span m="3079280">twice</span> <span m="3079730">n</span> <span m="3079910">minus</span> <span m="3080260">1</span> <span m="3080500">minus</span> <span m="3081070">n</span> <span m="3081200">minus</span> <span m="3081460">2,</span> <span m="3082050">is</span> <span m="3082150">what</span> <span m="3082270">we</span> <span m="3082380">would</span> <span m="3082510">do.</span> <span m="3083100">And</span> <span m="3083380">you</span> <span m="3083580">see that</span> <span m="3083750">it's</span> <span m="3083980">true.</span> <span m="3084790">So</span> <span m="3084970">you</span> <span m="3085230">could</span> <span m="3085360">really</span> <span m="3085680">prove</span> <span m="3086150">it</span> <span m="3086690">by</span> <span m="3087380">induction</span> <span m="3089000">for</span> <span m="3089180">any</span> <span m="3089390">of</span> <span m="3089490">the</span> <span m="3089520">solutions</span> <span m="3089920">we</span> <span m="3090030">ever</span> <span m="3090190">get.</span> <span m="3090720">And</span> <span m="3092090">if</span> <span m="3092210">you</span> <span m="3092340">worked</span> <span m="3092620">this</span> <span m="3092780">way</span> <span m="3092910">to</span> <span m="3092990">get</span> <span m="3093270">the</span> <span m="3093350">solution,</span> <span m="3093740">it</span> <span m="3093850">will</span> <span m="3094080">always</span> <span m="3094480">work.</span> <span m="3095390">This</span> <span m="3095580">method</span> <span m="3095890">never</span> <span m="3096270">fails,</span> <span m="3096850">gives</span> <span m="3097000">you</span> <span m="3097050">the</span> <span m="3097150">wrong</span> <span m="3097360">answer.</span> </p>
<p><span m="3098350">In</span> <span m="3098440">fact,</span> <span m="3099020">that's</span> <span m="3099120">an</span> <span m="3099180">important</span> <span m="3099470">point.</span> <span m="3099630">When we</span> <span m="3099820">tell</span> <span m="3099970">you</span> <span m="3100050">to</span> <span m="3100130">solve</span> <span m="3100560">a</span> <span m="3100630">recurrence</span> <span m="3101120">using</span> <span m="3101370">this</span> <span m="3101530">method</span> <span m="3101840">on</span> <span m="3102420">an</span> <span m="3102510">exam</span> <span m="3103480">or</span> <span m="3103720">for</span> <span m="3103860">homework,</span> <span m="3105100">you</span> <span m="3105270">don't</span> <span m="3105450">have</span> <span m="3105810">to</span> <span m="3105920">verify</span> <span m="3106205">it</span> <span m="3106490">by</span> <span m="3106690">induction</span> <span m="3107060">unless</span> <span m="3107350">we</span> <span m="3107460">say</span> <span m="3107880">verify</span> <span m="3108440">by</span> <span m="3108600">induction.</span> <span m="3109550">Because</span> <span m="3109850">we're</span> <span m="3109960">giving</span> <span m="3110310">you</span> <span m="3110470">a</span> <span m="3110570">method</span> <span m="3111130">that</span> <span m="3111470">is</span> <span m="3111660">guaranteed</span> <span m="3112180">to</span> <span m="3112260">work.</span> <span m="3112540">And if</span> <span m="3112660">you</span> <span m="3112730">do</span> <span m="3112840">the</span> <span m="3112940">method</span> <span m="3113230">right,</span> <span m="3113700">you're</span> <span m="3113840">fine.</span> <span m="3115220">There's</span> <span m="3115400">no</span> <span m="3116010">guessing</span> <span m="3116440">here</span> <span m="3116880">so</span> <span m="3117080">far.</span> <span m="3117960">In</span> <span m="3118090">a</span> <span m="3118140">minute,</span> <span m="3118460">we're</span> <span m="3118580">going</span> <span m="3118660">to</span> <span m="3118710">do</span> <span m="3118820">a</span> <span m="3118850">little</span> <span m="3119040">guessing,</span> <span m="3120440">but</span> <span m="3120830">not</span> <span m="3121060">so</span> <span m="3121230">far.</span> <span m="3123650">Any</span> <span m="3123770">other</span> <span m="3123940">questions?</span> <span m="3127670">OK.</span> </p>
<p><span m="3130100">This</span> <span m="3130450">works</span> <span m="3132000">for</span> <span m="3132340">all</span> <span m="3133520">homogeneous</span> <span m="3135150">linear</span> <span m="3135670">recurrences.</span> <span m="3137520">All right?</span> <span m="3137800">Remember</span> <span m="3138220">homogeneous</span> <span m="3138620">and</span> <span m="3139050">inhomogeneous</span> <span m="3139615">from</span> <span m="3139980">differential</span> <span m="3140430">equations?</span> <span m="3142450">It's</span> <span m="3142500">the</span> <span m="3142580">same</span> <span m="3142920">thing</span> <span m="3143190">happens</span> <span m="3143590">here.</span> <span m="3151600">And</span> <span m="3151760">now</span> <span m="3151970">we're</span> <span m="3152050">going</span> <span m="3152130">to</span> <span m="3152180">talk</span> <span m="3152440">about</span> <span m="3152690">the</span> <span m="3152870">inhomogeneous</span> <span m="3153560">case.</span> </p>
<p><span m="3162160">All</span> <span m="3162240">right.</span> <span m="3162450">So</span> <span m="3162570">we've</span> <span m="3162730">been</span> <span m="3162860">looking</span> <span m="3163230">at</span> <span m="3165710">linear</span> <span m="3167240">recurrences,</span> <span m="3167900">which</span> <span m="3168100">means</span> <span m="3168410">you</span> <span m="3168510">have</span> <span m="3168710">something</span> <span m="3169020">like</span> <span m="3169320">f</span> <span m="3169510">of</span> <span m="3169640">n</span> <span m="3170400">minus</span> <span m="3170960">a1</span> <span m="3172180">f</span> <span m="3172340">of</span> <span m="3172450">n</span> <span m="3172610">minus</span> <span m="3173000">1</span> <span m="3174450">minus</span> <span m="3175930">ad</span> <span m="3177056">f</span> <span m="3177440">of</span> <span m="3177540">n</span> <span m="3177680">minus</span> <span m="3178070">d</span> <span m="3179660">equals</span> <span m="3180480">0,</span> <span m="3181910">and</span> <span m="3182000">that</span> <span m="3182210">means</span> <span m="3182560">it's</span> <span m="3182810">homogeneous.</span> <span m="3189170">Now,</span> <span m="3189390">instead</span> <span m="3189670">of</span> <span m="3189740">0,</span> <span m="3190200">I</span> <span m="3190310">might</span> <span m="3190630">have</span> <span m="3190750">had</span> <span m="3191350">something</span> <span m="3191710">else</span> <span m="3191930">here,</span> <span m="3192430">might  have</span> <span m="3192640">been</span> <span m="3192800">equal</span> <span m="3193060">to</span> <span m="3193140">1,</span> <span m="3194930">maybe</span> <span m="3195310">n</span> <span m="3195480">squared,</span> <span m="3197620">or</span> <span m="3197870">some</span> <span m="3198130">general</span> <span m="3198660">function</span> <span m="3199130">g</span> <span m="3199360">of</span> <span m="3199450">n.</span> <span m="3201040">These</span> <span m="3201300">cases</span> <span m="3201730">are</span> <span m="3201880">all</span> <span m="3202610">inhomogeneous.</span> <span m="3210430">OK?</span> </p>
<p><span m="3211800">Now,</span> <span m="3213470">solving</span> <span m="3213950">inhomogeneous</span> <span m="3214820">linear</span> <span m="3215060">equations</span> <span m="3215570">is</span> <span m="3215680">just</span> <span m="3215900">one</span> <span m="3216120">step</span> <span m="3216380">harder</span> <span m="3216780">than</span> <span m="3217810">homogeneous.</span> <span m="3218450">Yeah?</span> </p>
<p><span m="3218750">AUDIENCE: All right.</span> <span m="3219241">Could you define</span> <span m="3219732">homogeneous</span> <span m="3220223">and</span> <span m="3220714">inhomogeneous</span> <span m="3221696">for people who</span> <span m="3222187">didn't take</span> <span m="3222678">differential</span> <span m="3223169">equations?</span> </p>
<p><span m="3223660">PROFESSOR: Yeah.</span> <span m="3224010">Homogeneous</span> <span m="3225560">means</span> <span m="3227270">you</span> <span m="3227400">have the</span> <span m="3227590">0</span> <span m="3227990">here.</span> <span m="3229170">You</span> <span m="3229280">got</span> <span m="3229420">all</span> <span m="3229620">these</span> <span m="3230350">f</span> <span m="3230590">terms</span> <span m="3231090">here,</span> <span m="3231570">right,</span> <span m="3232720">and</span> <span m="3233080">there's</span> <span m="3233270">nothing</span> <span m="3233630">else.</span> <span m="3234380">f of</span> <span m="3234670">n</span> <span m="3235260">minus</span> <span m="3235680">a1</span> <span m="3235790">f</span> <span m="3236000">n</span> <span m="3236230">minus</span> <span m="3236540">1</span> <span m="3237440">dot</span> <span m="3237570">dot</span> <span m="3237660">dot</span> <span m="3237940">minus</span> <span m="3238270">ad</span> <span m="3238540">f</span> <span m="3238670">n</span> <span m="3238720">minus</span> <span m="3239000">d</span> <span m="3239490">equals</span> <span m="3239880">0.</span> <span m="3241590">Like</span> <span m="3241870">with</span> <span m="3242960">Fibonacci</span> <span m="3243430">numbers--</span> <span m="3243980">f</span> <span m="3244180">of</span> <span m="3244300">n</span> <span m="3244560">minus</span> <span m="3245300">f n</span> <span m="3245610">minus</span> <span m="3245890">1</span> <span m="3246100">minus</span> <span m="3246420">f</span> <span m="3246550">n</span> <span m="3246650">minus</span> <span m="3246940">2</span> <span m="3247460">is</span> <span m="3247650">0.</span> <span m="3249640">OK?</span> </p>
<p><span m="3251250">I</span> <span m="3251370">could</span> <span m="3253280">consider</span> <span m="3253730">other</span> <span m="3254180">recurrences</span> <span m="3254710">where</span> <span m="3254820">there's</span> <span m="3255010">something</span> <span m="3255360">else</span> <span m="3255650">out</span> <span m="3255830">here that it</span> <span m="3256210">equals.</span> <span m="3257770">Like</span> <span m="3257960">I</span> <span m="3258010">could</span> <span m="3258240">have</span> <span m="3258430">a</span> <span m="3258500">Fibonacci</span> <span m="3259060">recurrence</span> <span m="3259550">where I</span> <span m="3259730">have</span> <span m="3260000">f(n)</span> <span m="3260680">equals</span> <span m="3261180">f</span> <span m="3261370">of</span> <span m="3261460">n</span> <span m="3261580">minus</span> <span m="3261840">1</span> <span m="3261960">plus</span> <span m="3262180">f(n)</span> <span m="3262420">minus</span> <span m="3262690">2</span> <span m="3263300">plus</span> <span m="3264410">g</span> <span m="3264620">of</span> <span m="3264700">n,</span> <span m="3264910">like</span> <span m="3265400">n</span> <span m="3265580">cubed.</span> <span m="3266780">That</span> <span m="3266940">would</span> <span m="3267060">give</span> <span m="3267200">me</span> <span m="3267420">a</span> <span m="3267940">Fibonacci-like</span> <span m="3268730">recurrence.</span> <span m="3269820">And</span> <span m="3269940">as</span> <span m="3270040">soon</span> <span m="3270250">as</span> <span m="3270330">you</span> <span m="3270430">put</span> <span m="3270620">a</span> <span m="3270690">non-zero</span> <span m="3271380">out</span> <span m="3271540">here,</span> <span m="3272680">then</span> <span m="3272960">it's</span> <span m="3273160">inhomogeneous.</span> <span m="3274980">And now</span> <span m="3275210">I'm</span> <span m="3275370">going</span> <span m="3275480">to</span> <span m="3275790">tell</span> <span m="3275950">you</span> <span m="3276030">how</span> <span m="3276150">to</span> <span m="3276240">solve</span> <span m="3276900">that</span> <span m="3277320">kind</span> <span m="3277490">of</span> <span m="3277540">recurrence,</span> <span m="3277960">which</span> <span m="3278160">is</span> <span m="3278720">more</span> <span m="3278870">general.</span> <span m="3281670">OK?</span> <span m="3284660">OK.</span> </p>
<p><span m="3289000">So</span> <span m="3289200">let</span> <span m="3289320">me</span> <span m="3289450">outline</span> <span m="3289860">the</span> <span m="3289940">method,</span> <span m="3290370">and</span> <span m="3290450">then</span> <span m="3290580">we'll</span> <span m="3290680">do</span> <span m="3290810">an</span> <span m="3290870">example.</span> <span m="3302220">All</span> <span m="3302550">right.</span> <span m="3302840">So</span> <span m="3305430">the</span> <span m="3305530">general</span> <span m="3307340">inhomogeneous</span> <span m="3308240">recurrence</span> <span m="3322140">is</span> <span m="3322510">exactly</span> <span m="3323000">this.</span> <span m="3323330">It's</span> <span m="3323520">f</span> <span m="3323730">of</span> <span m="3323840">n</span> <span m="3325310">minus</span> <span m="3325585">a1</span> <span m="3327190">f of n</span> <span m="3327540">minus</span> <span m="3327910">1</span> <span m="3328850">minus</span> <span m="3330250">a</span> <span m="3330630">sub</span> <span m="3330740">d</span> <span m="3331592">f</span> <span m="3332020">of</span> <span m="3332130">n</span> <span m="3332280">minus</span> <span m="3332620">d</span> <span m="3334110">equals</span> <span m="3334970">g</span> <span m="3335210">of</span> <span m="3335290">n,</span> <span m="3336320">where</span> <span m="3336420">that's</span> <span m="3336660">some</span> <span m="3336890">fixed</span> <span m="3337790">function</span> <span m="3338270">of</span> <span m="3338360">n,</span> <span m="3338890">nothing</span> <span m="3339130">to</span> <span m="3339200">do</span> <span m="3339350">with</span> <span m="3339410">f.</span> </p>
<p><span m="3342550">And</span> <span m="3342720">we</span> <span m="3342880">solve</span> <span m="3343370">it</span> <span m="3344140">according</span> <span m="3344480">to</span> <span m="3344520">the</span> <span m="3344620">following</span> <span m="3345110">three-step</span> <span m="3345850">method.</span> <span m="3349200">In</span> <span m="3349460">step</span> <span m="3349820">1,</span> <span m="3351370">we</span> <span m="3351500">replace</span> <span m="3354690">g</span> <span m="3354970">of</span> <span m="3355040">n</span> <span m="3356860">by</span> <span m="3357030">0,</span> <span m="3359280">thereby</span> <span m="3359700">creating</span> <span m="3361640">the</span> <span m="3361740">situation</span> <span m="3362290">we</span> <span m="3362380">already  know</span> <span m="3362510">how to</span> <span m="3362987">solve.</span> <span m="3366330">And</span> <span m="3366450">we</span> <span m="3366560">solve</span> <span m="3369320">the</span> <span m="3369800">homogeneous</span> <span m="3370820">recurrence--</span> <span m="3371450">because</span> <span m="3371580">you</span> <span m="3371660">got</span> <span m="3371760">a</span> <span m="3371790">0</span> <span m="3372130">there</span> <span m="3372450">now--</span> <span m="3381820">but</span> <span m="3382120">we</span> <span m="3382320">don't</span> <span m="3382580">go</span> <span m="3382690">all</span> <span m="3382950">the</span> <span m="3383040">way.</span> <span m="3385220">We</span> <span m="3385460">ignore</span> <span m="3385990">the</span> <span m="3386100">boundary</span> <span m="3386480">conditions</span> <span m="3387310">for</span> <span m="3387410">now.</span> <span m="3395500">So</span> <span m="3395720">you don't</span> <span m="3395860">get</span> <span m="3395980">all</span> <span m="3396230">the</span> <span m="3396320">way</span> <span m="3396460">to</span> <span m="3396520">the</span> <span m="3396620">final</span> <span m="3396970">answer</span> <span m="3397320">where</span> <span m="3397460">you</span> <span m="3397600">plugged</span> <span m="3397955">in</span> <span m="3398540">the</span> <span m="3398610">boundary</span> <span m="3398950">conditions</span> <span m="3399420">to</span> <span m="3399490">get</span> <span m="3399640">the</span> <span m="3399710">constant</span> <span m="3400370">coefficients.</span> <span m="3401510">Leave</span> <span m="3401900">the</span> <span m="3401980">constant</span> <span m="3402400">coefficients</span> <span m="3403010">undecided.</span> </p>
<p><span m="3405510">Then in</span> <span m="3405730">step</span> <span m="3406070">2,</span> <span m="3409060">we</span> <span m="3409210">put</span> <span m="3409470">back</span> <span m="3409740">in</span> <span m="3409880">g of</span> <span m="3410170">n,</span> <span m="3415420">and</span> <span m="3415570">we</span> <span m="3415710">find</span> <span m="3418140">any</span> <span m="3418540">what's</span> <span m="3418750">called</span> <span m="3419200">particular</span> <span m="3419810">solution,</span> <span m="3425280">again</span> <span m="3425880">ignoring</span> <span m="3426300">boundary</span> <span m="3426630">conditions.</span> <span m="3428126">So</span> <span m="3428550">you'll</span> <span m="3428770">have the</span> <span m="3429050">constant</span> <span m="3429460">factors</span> <span m="3429900">there too.</span> <span m="3440240">All</span> <span m="3440330">right?</span> <span m="3440640">And</span> <span m="3440740">then</span> <span m="3440930">step</span> <span m="3441280">3</span> <span m="3442600">is</span> <span m="3442770">going</span> <span m="3442890">to</span> <span m="3442960">be</span> <span m="3443170">to</span> <span m="3443840">put</span> <span m="3444030">the</span> <span m="3444100">whole</span> <span m="3444350">thing</span> <span m="3444550">together</span> <span m="3444990">and</span> <span m="3445120">plug in</span> <span m="3445450">the</span> <span m="3445520">boundary</span> <span m="3445880">conditions</span> <span m="3446377">and</span> <span m="3446874">solve it.</span> </p>
<p><span m="3455330">So</span> <span m="3455490">we</span> <span m="3455740">add</span> <span m="3457110">the</span> <span m="3457430">homogeneous</span> <span m="3459910">and</span> <span m="3460220">particular--</span> <span m="3461675">and</span> <span m="3462170">I'll</span> <span m="3462300">show</span> <span m="3462470">you</span> <span m="3462540">how</span> <span m="3462690">to</span> <span m="3462770">do</span> <span m="3462890">step</span> <span m="3463120">2</span> <span m="3463240">in</span> <span m="3463320">a</span> <span m="3463360">minute--</span> <span m="3474420">so</span> <span m="3474920">we</span> <span m="3475060">add</span> <span m="3475260">the</span> <span m="3475330">homogeneous</span> <span m="3475870">and</span> <span m="3475960">particular</span> <span m="3476440">solutions</span> <span m="3476910">together</span> <span m="3485930">and</span> <span m="3486190">then</span> <span m="3486490">use</span> <span m="3486790">the</span> <span m="3486870">boundary</span> <span m="3487250">conditions</span> <span m="3487810">to</span> <span m="3487890">get</span> <span m="3488050">the</span> <span m="3488170">answer.</span> </p>
<p><span m="3510740">OK.</span> <span m="3511060">So</span> <span m="3511560">let's</span> <span m="3511800">do</span> <span m="3511940">an</span> <span m="3512010">example.</span> <span m="3515750">Let's</span> <span m="3516040">go</span> <span m="3516200">down</span> <span m="3516380">here</span> <span m="3516590">and</span> <span m="3516670">do</span> <span m="3516790">an</span> <span m="3516870">example</span> <span m="3517320">of</span> <span m="3517390">the</span> <span m="3517530">three</span> <span m="3517880">steps.</span> <span m="3537480">So</span> <span m="3537750">let's</span> <span m="3538060">say</span> <span m="3538310">that our</span> <span m="3538450">recurrence</span> <span m="3539170">is</span> <span m="3540780">this</span> <span m="3542290">nasty-looking</span> <span m="3543000">thing--</span> <span m="3544090">f</span> <span m="3544310">of</span> <span m="3544420">n</span> <span m="3545330">equals</span> <span m="3545850">4</span> <span m="3547110">f</span> <span m="3547290">of</span> <span m="3547400">n</span> <span m="3547560">minus</span> <span m="3547930">1</span> <span m="3549190">plus</span> <span m="3550380">3</span> <span m="3550950">to</span> <span m="3551010">the</span> <span m="3551180">n.</span> <span m="3552850">And</span> <span m="3553180">the</span> <span m="3553250">boundary</span> <span m="3553650">condition</span> <span m="3554170">is that</span> <span m="3554670">f</span> <span m="3554890">of</span> <span m="3554990">1</span> <span m="3556420">is</span> <span m="3556650">1.</span> </p>
<p><span m="3559240">So</span> <span m="3559590">step</span> <span m="3559950">1</span> <span m="3560590">says,</span> <span m="3562320">ignore</span> <span m="3562870">this</span> <span m="3563150">3</span> <span m="3563430">to</span> <span m="3563500">the</span> <span m="3563660">n</span> <span m="3563870">thing</span> <span m="3564800">and</span> <span m="3564940">get</span> <span m="3565120">back</span> <span m="3565390">to</span> <span m="3565540">just</span> <span m="3565820">the</span> <span m="3565870">homogeneous</span> <span m="3566570">form,</span> <span m="3567770">which</span> <span m="3568130">is</span> <span m="3568400">f</span> <span m="3568620">of</span> <span m="3568760">n</span> <span m="3569080">minus</span> <span m="3569630">4</span> <span m="3570100">f</span> <span m="3570280">of</span> <span m="3570370">n</span> <span m="3570490">minus</span> <span m="3570830">1</span> <span m="3571030">is</span> <span m="3571150">0.</span> <span m="3572490">So</span> <span m="3572790">let's</span> <span m="3573110">solve</span> <span m="3573470">that.</span> <span m="3573720">What's</span> <span m="3573990">the</span> <span m="3574260">characteristic</span> <span m="3574880">equation</span> <span m="3576720">for</span> <span m="3577820">the</span> <span m="3577890">homogeneous</span> <span m="3578470">part?</span> <span m="3581240">What's</span> <span m="3581430">the</span> <span m="3581520">characteristic</span> <span m="3582020">equation?</span> <span m="3585789">What is it?</span> </p>
<p><span m="3586723">AUDIENCE: Alpha</span> <span m="3587190">[INAUDIBLE].</span> </p>
<p><span m="3589060">PROFESSOR: Close.</span> <span m="3589720">It's--</span> <span m="3590540">you</span> <span m="3590690">could</span> <span m="3590810">either</span> <span m="3591000">say</span> <span m="3591310">alpha to the</span> <span m="3591590">n</span> <span m="3591940">minus</span> <span m="3592350">4</span> <span m="3592770">alpha to the</span> <span m="3593080">n</span> <span m="3593360">minus</span> <span m="3593740">1.</span> <span m="3594780">But</span> <span m="3594890">better</span> <span m="3595200">to</span> <span m="3595300">simplify</span> <span m="3595990">it</span> <span m="3596040">into</span> <span m="3596300">an</span> <span m="3596380">order</span> <span m="3596620">1</span> <span m="3597550">polynomial.</span> <span m="3599080">So</span> <span m="3599410">it would</span> <span m="3599530">be</span> <span m="3599970">alpha</span> <span m="3600420">minus</span> <span m="3600830">4,</span> <span m="3602090">really</span> <span m="3602390">simple</span> <span m="3603210">in</span> <span m="3603250">this</span> <span m="3603410">case.</span> <span m="3604470">So</span> <span m="3604590">the</span> <span m="3604700">characteristic</span> <span m="3605200">equation</span> <span m="3605590">is</span> <span m="3605710">alpha</span> <span m="3606030">minus</span> <span m="3606360">4,</span> <span m="3606950">alpha</span> <span m="3608040">minus</span> <span m="3608530">4,</span> <span m="3609666">all</span> <span m="3610020">right,</span> <span m="3610710">equals</span> <span m="3611040">0.</span> </p>
<p><span m="3612850">And</span> <span m="3613190">it's</span> <span m="3613330">really</span> <span m="3613780">easy</span> <span m="3614090">to</span> <span m="3614190">find</span> <span m="3614520">the</span> <span m="3614610">root</span> <span m="3614890">of</span> <span m="3614960">this</span> <span m="3615170">thing.</span> <span m="3616600">It's</span> <span m="3616740">just</span> <span m="3616980">alpha</span> <span m="3617300">equals</span> <span m="3617690">4</span> <span m="3618740">is</span> <span m="3619100">your</span> <span m="3619250">only</span> <span m="3619510">root.</span> <span m="3620840">And</span> <span m="3620940">that</span> <span m="3621130">means</span> <span m="3621490">the</span> <span m="3622170">homogeneous</span> <span m="3622910">solution</span> <span m="3627040">is</span> <span m="3628920">f</span> <span m="3629140">of</span> <span m="3629280">n</span> <span m="3630120">equals</span> <span m="3630440">a</span> <span m="3630510">constant</span> <span m="3631240">times</span> <span m="3631660">4 to the n.</span> <span m="3634980">All</span> <span m="3635080">right?</span> <span m="3635260">So</span> <span m="3635340">that's</span> <span m="3635560">step</span> <span m="3635840">1.</span> </p>
<p><span m="3643670">OK.</span> <span m="3644360">Let's</span> <span m="3644800">do</span> <span m="3645670">step</span> <span m="3646170">2,</span> <span m="3646790">which</span> <span m="3646940">we've</span> <span m="3647100">not</span> <span m="3647620">tried</span> <span m="3647890">before.</span> <span m="3651910">We</span> <span m="3652050">need</span> <span m="3652210">to</span> <span m="3652270">find</span> <span m="3652680">a</span> <span m="3652740">particular</span> <span m="3653380">solution,</span> <span m="3656830">and</span> <span m="3657300">this</span> <span m="3657510">just</span> <span m="3657740">means</span> <span m="3658020">any</span> <span m="3658310">old</span> <span m="3658520">solution</span> <span m="3664520">to</span> <span m="3665630">f</span> <span m="3665870">of</span> <span m="3665990">n</span> <span m="3667160">minus</span> <span m="3667690">4</span> <span m="3668330">f</span> <span m="3668540">of</span> <span m="3668670">n</span> <span m="3668830">minus</span> <span m="3669190">1</span> <span m="3670480">equals</span> <span m="3670900">3</span> <span m="3671140">to</span> <span m="3671190">the</span> <span m="3671350">n,</span> <span m="3673590">without</span> <span m="3673910">worrying</span> <span m="3674130">about</span> <span m="3674350">boundary</span> <span m="3674640">conditions.</span> </p>
<p><span m="3675970">Now,</span> <span m="3677800">there's</span> <span m="3678400">basic</span> <span m="3679130">rules</span> <span m="3679830">to</span> <span m="3679950">use</span> <span m="3680430">to</span> <span m="3680530">figure</span> <span m="3680880">out</span> <span m="3681395">what</span> <span m="3681760">to</span> <span m="3681840">guess</span> <span m="3682180">here.</span> <span m="3683690">And</span> <span m="3684700">basically,</span> <span m="3685240">the</span> <span m="3685400">idea</span> <span m="3686670">is</span> <span m="3687020">that</span> <span m="3687240">you</span> <span m="3687360">guess</span> <span m="3687800">something</span> <span m="3688780">for</span> <span m="3688940">f</span> <span m="3689120">of</span> <span m="3689250">n</span> <span m="3690210">that</span> <span m="3690360">looks</span> <span m="3690860">a</span> <span m="3690930">whole</span> <span m="3691210">lot</span> <span m="3691500">like</span> <span m="3692320">this</span> <span m="3692550">g</span> <span m="3692820">term</span> <span m="3693100">out</span> <span m="3693230">here.</span> <span m="3694800">And</span> <span m="3694960">in</span> <span m="3695020">particular, if</span> <span m="3695520">this</span> <span m="3695740">g</span> <span m="3696000">term</span> <span m="3696260">is</span> <span m="3696410">3</span> <span m="3696630">to</span> <span m="3696690">the</span> <span m="3696850">n,</span> <span m="3698070">you</span> <span m="3698260">guess</span> <span m="3698740">a</span> <span m="3698820">constant</span> <span m="3699380">times</span> <span m="3699620">3</span> <span m="3699790">to</span> <span m="3699840">the</span> <span m="3700000">n.</span> <span m="3701270">If</span> <span m="3701480">it's</span> <span m="3701740">5 to the</span> <span m="3702040">n,</span> <span m="3702470">you</span> <span m="3702580">guess</span> <span m="3702850">a</span> <span m="3702920">constant</span> <span m="3703330">times</span> <span m="3703590">5</span> <span m="3703690">to</span> <span m="3703920">the</span> <span m="3704050">n.</span> <span m="3704680">If</span> <span m="3704840">it's</span> <span m="3705010">n</span> <span m="3705200">squared,</span> <span m="3705800">you</span> <span m="3705930">guess</span> <span m="3706360">a</span> <span m="3706620">polynomial</span> <span m="3707410">that's</span> <span m="3707780">of</span> <span m="3707900">degree</span> <span m="3708210">2</span> <span m="3709420">and</span> <span m="3709620">so</span> <span m="3709810">forth.</span> <span m="3711610">OK?</span> </p>
<p><span m="3712090">So</span> <span m="3713070">let's</span> <span m="3713450">guess</span> <span m="3714590">a</span> <span m="3714690">constant</span> <span m="3715270">times 3</span> <span m="3715530">to the n</span> <span m="3718920">and</span> <span m="3719120">see</span> <span m="3719280">if</span> <span m="3719380">we</span> <span m="3719470">can</span> <span m="3719570">make</span> <span m="3719750">it</span> <span m="3719850">work,</span> <span m="3721290">not</span> <span m="3721560">worrying</span> <span m="3721800">about</span> <span m="3722150">the</span> <span m="3722460">boundary</span> <span m="3722956">conditions.</span> <span m="3733890">All</span> <span m="3734060">right.</span> <span m="3736346">And</span> <span m="3736760">this</span> <span m="3737040">is</span> <span m="3737170">just</span> <span m="3737430">like</span> <span m="3737570">differential</span> <span m="3738070">equations,</span> <span m="3738630">right?</span> <span m="3740220">Same</span> <span m="3740850">guessing</span> <span m="3741200">strategy</span> <span m="3741730">exactly</span> <span m="3742065">you</span> <span m="3742400">use in</span> <span m="3742690">differential</span> <span m="3743170">equations,</span> <span m="3743650">those</span> <span m="3743850">of</span> <span m="3743930">you</span> <span m="3744570">who've</span> <span m="3744690">had</span> <span m="3744910">that.</span> </p>
<p><span m="3746250">So</span> <span m="3746380">we</span> <span m="3746510">guess</span> <span m="3747620">f</span> <span m="3747840">of</span> <span m="3747960">n</span> <span m="3748420">equals</span> <span m="3748780">a</span> <span m="3748840">constant</span> <span m="3749330">times</span> <span m="3749580">3</span> <span m="3749780">to</span> <span m="3749830">the</span> <span m="3749970">n,</span> <span m="3751840">and</span> <span m="3752010">let's</span> <span m="3752200">plug</span> <span m="3752420">it</span> <span m="3752480">in.</span> <span m="3754320">So</span> <span m="3755890">we</span> <span m="3755950">plug</span> <span m="3756350">that</span> <span m="3756590">in</span> <span m="3756940">up</span> <span m="3757330">there.</span> <span m="3757640">We</span> <span m="3757750">get</span> <span m="3759000">c3</span> <span m="3759950">to</span> <span m="3760010">the</span> <span m="3760180">n</span> <span m="3760760">minus</span> <span m="3761220">c3</span> <span m="3761820">to</span> <span m="3761890">the</span> <span m="3762040">n</span> <span m="3762180">minus</span> <span m="3762590">1</span> <span m="3764300">equals</span> <span m="3765730">3</span> <span m="3766100">to</span> <span m="3766150">the</span> <span m="3766280">n.</span> <span m="3767705">All right?</span> <span m="3768100">So</span> <span m="3768480">let's</span> <span m="3768610">divide</span> <span m="3769140">by</span> <span m="3769410">3</span> <span m="3769660">to</span> <span m="3769720">the</span> <span m="3769860">n</span> <span m="3770010">here.</span> <span m="3770240">We</span> <span m="3770350">get</span> <span m="3771400">3c</span> <span m="3772730">minus</span> <span m="3773260">c</span> <span m="3774790">equals</span> <span m="3775290">3.</span> <span m="3776710">Did I</span> <span m="3776830">do</span> <span m="3776950">that</span> <span m="3777140">right?</span> <span m="3778570">No,</span> <span m="3779050">I</span> <span m="3779170">left</span> <span m="3779470">off</span> <span m="3779630">my</span> <span m="3779760">4</span> <span m="3780120">here,</span> <span m="3780350">right,</span> <span m="3781222">left off</span> <span m="3781660">that. So</span> <span m="3781970">there's a</span> <span m="3782160">4</span> <span m="3782580">here.</span> <span m="3786150">And</span> <span m="3786290">that</span> <span m="3786520">means</span> <span m="3786890">that</span> <span m="3788130">c</span> <span m="3788500">is</span> <span m="3788640">minus</span> <span m="3789010">3.</span> <span m="3791150">OK?</span> <span m="3791400">So</span> <span m="3791500">I</span> <span m="3791540">got</span> <span m="3791640">minus</span> <span m="3792110">c</span> <span m="3792320">equals</span> <span m="3792670">3,</span> <span m="3793050">so</span> <span m="3793190">c</span> <span m="3793480">is</span> <span m="3793630">minus</span> <span m="3794080">3.</span> </p>
<p><span m="3795890">All right.</span> <span m="3796100">That</span> <span m="3796370">means that</span> <span m="3796680">the</span> <span m="3796780">particular</span> <span m="3797320">solution</span> <span m="3804980">is</span> <span m="3805230">just</span> <span m="3805650">f</span> <span m="3805950">of</span> <span m="3806070">n</span> <span m="3807860">equals--</span> <span m="3810360">c</span> <span m="3810520">is</span> <span m="3810650">minus</span> <span m="3811080">3--</span> <span m="3812550">minus</span> <span m="3813240">3</span> <span m="3813490">to</span> <span m="3813550">the</span> <span m="3813680">n</span> <span m="3813830">plus</span> <span m="3814090">1.</span> <span m="3816270">All right?</span> <span m="3816560">So now</span> <span m="3816700">I</span> <span m="3816810">found</span> <span m="3817420">a</span> <span m="3817560">solution</span> <span m="3818970">where</span> <span m="3819120">there's</span> <span m="3819290">no</span> <span m="3819460">constants</span> <span m="3820020">involved.</span> <span m="3821370">This</span> <span m="3821590">time,</span> <span m="3822270">the</span> <span m="3822360">constant</span> <span m="3822820">went</span> <span m="3822990">away</span> <span m="3823430">just</span> <span m="3823780">plugging</span> <span m="3824170">into</span> <span m="3825920">the</span> <span m="3826030">recurrence</span> <span m="3826520">formula.</span> <span m="3826830">I</span> <span m="3826900">didn't</span> <span m="3827190">use</span> <span m="3827410">base</span> <span m="3827700">cases</span> <span m="3828050">yet.</span> <span m="3829130">I</span> <span m="3829230">just</span> <span m="3829480">found</span> <span m="3829700">a</span> <span m="3829760">particular</span> <span m="3830470">case</span> <span m="3830900">that</span> <span m="3831050">works</span> <span m="3831460">for</span> <span m="3831550">the</span> <span m="3831670">recurrence--</span> <span m="3833230">minus</span> <span m="3833610">3</span> <span m="3833790">to</span> <span m="3833840">the</span> <span m="3833940">n</span> <span m="3834060">plus</span> <span m="3834300">1.</span> <span m="3836980">OK?</span> <span m="3839492">All right.</span> </p>
<p><span m="3841430">Now</span> <span m="3841600">we</span> <span m="3841700">go</span> <span m="3841790">to</span> <span m="3841880">step</span> <span m="3842050">3.</span> <span m="3859430">All</span> <span m="3859570">right.</span> <span m="3859800">Step</span> <span m="3860100">3,</span> <span m="3863870">we</span> <span m="3864000">find</span> <span m="3864280">the</span> <span m="3864360">general</span> <span m="3864830">solution</span> <span m="3870670">by</span> <span m="3870910">adding</span> <span m="3871520">the</span> <span m="3871630">homogeneous</span> <span m="3872420">and</span> <span m="3872500">the</span> <span m="3872590">particular</span> <span m="3873040">solution.</span> <span m="3877850">So</span> <span m="3877950">we</span> <span m="3878070">take</span> <span m="3878340">f</span> <span m="3878530">of</span> <span m="3878640">n</span> <span m="3880080">equals</span> <span m="3882660">c1</span> <span m="3883550">times</span> <span m="3883810">4</span> <span m="3884000">to</span> <span m="3884040">the</span> <span m="3884180">n</span> <span m="3888070">plus</span> <span m="3889040">negative</span> <span m="3889870">3</span> <span m="3890090">to</span> <span m="3890130">the</span> <span m="3890250">n</span> <span m="3890390">plus</span> <span m="3890610">1.</span> <span m="3895350">And</span> <span m="3895490">all</span> <span m="3895760">I've</span> <span m="3895870">got</span> <span m="3896050">to</span> <span m="3896140">do</span> <span m="3896410">is</span> <span m="3896490">determine</span> <span m="3896890">c1,</span> <span m="3898660">and</span> <span m="3898830">I</span> <span m="3898910">do</span> <span m="3899100">that</span> <span m="3899510">from</span> <span m="3899680">the</span> <span m="3899770">boundary</span> <span m="3900260">conditions.</span> </p>
<p><span m="3902090">Yeah, let's</span> <span m="3902470">see.</span> <span m="3902560">Where</span> <span m="3902800">was</span> <span m="3902910">my</span> <span m="3903050">boundary</span> <span m="3903450">condition</span> <span m="3903890">here?</span> <span m="3905800">Ah, up</span> <span m="3906290">there--</span> <span m="3906860">f</span> <span m="3907110">of</span> <span m="3907200">1</span> <span m="3907470">is</span> <span m="3907610">1.</span> <span m="3909580">So</span> <span m="3909870">f</span> <span m="3910110">of</span> <span m="3910210">1</span> <span m="3911330">equals</span> <span m="3911670">1.</span> <span m="3912690">Plugging</span> <span m="3913120">in</span> <span m="3913200">1</span> <span m="3913510">here,</span> <span m="3913780">I</span> <span m="3913840">get</span> <span m="3914070">c1</span> <span m="3914700">times</span> <span m="3915120">4</span> <span m="3915760">to</span> <span m="3915810">the</span> <span m="3915930">1</span> <span m="3917292">minus</span> <span m="3918200">3</span> <span m="3918900">squared.</span> <span m="3921640">All</span> <span m="3921740">right?</span> <span m="3921930">So</span> <span m="3922090">I</span> <span m="3922180">have</span> <span m="3922520">4c1</span> <span m="3923950">minus</span> <span m="3924290">9.</span> <span m="3925120">Put</span> <span m="3925290">the</span> <span m="3925370">9</span> <span m="3925720">over</span> <span m="3925930">here,</span> <span m="3927060">I</span> <span m="3927190">get</span> <span m="3927430">10</span> <span m="3927880">equals</span> <span m="3928250">4c1.</span> <span m="3930100">And</span> <span m="3930200">that</span> <span m="3930430">means</span> <span m="3930800">that</span> <span m="3931570">c1</span> <span m="3932240">is</span> <span m="3932540">5/2.</span> <span m="3937570">And</span> <span m="3937640">now</span> <span m="3937890">I</span> <span m="3937980">know</span> <span m="3938230">the</span> <span m="3938410">final</span> <span m="3938850">solution</span> <span m="3939360">for</span> <span m="3939540">f(n).</span> <span m="3944170">It's</span> <span m="3944440">5/2</span> <span m="3947060">4 to</span> <span m="3947300">the</span> <span m="3947510">n</span> <span m="3948230">minus</span> <span m="3948680">3 to the</span> <span m="3949060">n</span> <span m="3949250">plus</span> <span m="3949520">1.</span> </p>
<p><span m="3956650">Now,</span> <span m="3956880">when</span> <span m="3956990">you're</span> <span m="3957170">all</span> <span m="3957390">done</span> <span m="3957610">with</span> <span m="3957750">this,</span> <span m="3958400">you</span> <span m="3958520">don't</span> <span m="3958800">have--</span> <span m="3959130">we</span> <span m="3959240">won't</span> <span m="3959380">make</span> <span m="3959590">you</span> <span m="3959660">do an</span> <span m="3959930">inductive</span> <span m="3960380">proof,</span> <span m="3960830">which</span> <span m="3961000">you</span> <span m="3961100">could</span> <span m="3961340">do</span> <span m="3962570">to</span> <span m="3962730">verify</span> <span m="3963180">it's</span> <span m="3963310">right.</span> <span m="3963880">If</span> <span m="3963970">you</span> <span m="3964050">wanted</span> <span m="3964140">to</span> <span m="3964180">be</span> <span m="3964280">really,</span> <span m="3964830">really</span> <span m="3965020">careful,</span> <span m="3965370">you</span> <span m="3965490">should</span> <span m="3965770">do</span> <span m="3965880">that.</span> <span m="3966950">But</span> <span m="3967110">it</span> <span m="3967260">is</span> <span m="3967540">a</span> <span m="3967620">good</span> <span m="3967840">idea</span> <span m="3968970">just</span> <span m="3969210">to</span> <span m="3969300">check</span> <span m="3969540">a</span> <span m="3969630">couple</span> <span m="3969930">values</span> <span m="3970310">of</span> <span m="3970430">n</span> <span m="3971360">to</span> <span m="3971490">make</span> <span m="3971670">sure</span> <span m="3971930">you</span> <span m="3972000">didn't</span> <span m="3972170">make</span> <span m="3972350">a</span> <span m="3972400">mistake</span> <span m="3972960">because</span> <span m="3973130">you</span> <span m="3973230">might</span> <span m="3973450">have. With</span> <span m="3973620">all</span> <span m="3973850">these</span> <span m="3974040">calculations,</span> <span m="3974780">you</span> <span m="3974850">might</span> <span m="3974990">have</span> <span m="3975060">made</span> <span m="3975190">a</span> <span m="3975230">mistake.</span> </p>
<p><span m="3976440">So</span> <span m="3976760">let's</span> <span m="3977030">check,</span> <span m="3978110">for</span> <span m="3978250">example,</span> <span m="3979650">f</span> <span m="3979760">of</span> <span m="3979870">2.</span> <span m="3983000">So</span> <span m="3984240">by</span> <span m="3984470">the</span> <span m="3984590">recurrence,</span> <span m="3986780">f</span> <span m="3987040">of</span> <span m="3987190">2</span> <span m="3989070">is</span> <span m="3989340">4</span> <span m="3989940">times</span> <span m="3990300">f</span> <span m="3990520">of</span> <span m="3990650">1--</span> <span m="3991815">it's</span> <span m="3992310">4</span> <span m="3992510">times</span> <span m="3992810">1--</span> <span m="3993760">plus</span> <span m="3994960">3</span> <span m="3995270">squared,</span> <span m="3996320">which</span> <span m="3996360">is</span> <span m="3996500">9,</span> <span m="3997880">is</span> <span m="3998180">13.</span> <span m="3999960">And</span> <span m="4000060">let's</span> <span m="4000170">just</span> <span m="4000410">check</span> <span m="4000720">when we</span> <span m="4000860">plug</span> <span m="4001160">in 2</span> <span m="4001650">here.</span> <span m="4003790">f</span> <span m="4004060">of</span> <span m="4004200">2</span> <span m="4005510">is</span> <span m="4005770">5/2</span> <span m="4007390">times</span> <span m="4007630">16</span> <span m="4009100">minus</span> <span m="4009770">27.</span> <span m="4012590">5</span> <span m="4012900">times</span> <span m="4013180">8</span> <span m="4013320">is</span> <span m="4013460">40,</span> <span m="4013820">minus</span> <span m="4014170">27</span> <span m="4014810">is</span> <span m="4014930">13.</span> <span m="4017210">Just</span> <span m="4017560">as</span> <span m="4017670">a</span> <span m="4017750">sanity</span> <span m="4018190">check.</span> <span m="4019230">Because</span> <span m="4019490">there's</span> <span m="4019630">a</span> <span m="4019700">decent</span> <span m="4020080">chance</span> <span m="4020340">if</span> <span m="4020390">you</span> <span m="4020500">made</span> <span m="4020720">a</span> <span m="4020760">mistake,</span> <span m="4021170">it'll</span> <span m="4021360">catch</span> <span m="4021650">it</span> <span m="4021730">pretty</span> <span m="4021980">quick</span> <span m="4022280">with</span> <span m="4022430">n</span> <span m="4022550">equals</span> <span m="4022800">2</span> <span m="4022980">or</span> <span m="4023060">3.</span> <span m="4024540">All right?</span> <span m="4024740">Just to</span> <span m="4025040">make</span> <span m="4025180">sure</span> <span m="4025340">you</span> <span m="4025430">got</span> <span m="4025570">it</span> <span m="4025620">right.</span> <span m="4027880">Any</span> <span m="4028375">questions</span> <span m="4031850">about</span> <span m="4032070">that?</span> </p>
<p><span m="4034280">So</span> <span m="4034410">the</span> <span m="4034510">tricky</span> <span m="4034830">part</span> <span m="4035160">here</span> <span m="4035820">is</span> <span m="4036070">guessing</span> <span m="4036630">the</span> <span m="4036730">particular</span> <span m="4037260">solution.</span> <span m="4038730">So</span> <span m="4038770">let</span> <span m="4038880">me</span> <span m="4038990">give</span> <span m="4039140">you</span> <span m="4039180">the</span> <span m="4039340">rules</span> <span m="4039850">for</span> <span m="4039990">that,</span> <span m="4040980">just</span> <span m="4041120">write</span> <span m="4041350">those</span> <span m="4041550">down,</span> <span m="4044250">and</span> <span m="4044680">maybe</span> <span m="4044910">I'll</span> <span m="4045060">do</span> <span m="4045390">one</span> <span m="4045540">last</span> <span m="4045850">example.</span> <span m="4060520">All right.</span> <span m="4060820">So</span> <span m="4062116">we're</span> <span m="4062550">guessing</span> <span m="4063000">a</span> <span m="4063060">particular</span> <span m="4063600">solution.</span> <span m="4076740">So</span> <span m="4078190">if</span> <span m="4078470">g--</span> <span m="4079650">that's</span> <span m="4079930">saying</span> <span m="4080460">the</span> <span m="4080560">non-homogeneous</span> <span m="4081470">part--</span> <span m="4083570">is</span> <span m="4084210">exponential,</span> <span m="4091130">you</span> <span m="4091250">should</span> <span m="4091430">try</span> <span m="4091720">guessing</span> <span m="4094130">an</span> <span m="4094330">exponential</span> <span m="4094660">of</span> <span m="4094990">the</span> <span m="4095080">same</span> <span m="4095390">type.</span> </p>
<p><span m="4100740">So</span> <span m="4100840">for</span> <span m="4100970">example,</span> <span m="4103399">say</span> <span m="4103729">that</span> <span m="4104029">g</span> <span m="4106250">is</span> <span m="4107189">2 to</span> <span m="4107380">the</span> <span m="4107600">n</span> <span m="4108060">plus</span> <span m="4108380">3</span> <span m="4108600">to</span> <span m="4108660">the</span> <span m="4108830">n.</span> <span m="4111050">What</span> <span m="4111270">you</span> <span m="4111359">should</span> <span m="4111580">do</span> <span m="4111939">is</span> <span m="4112130">guess</span> <span m="4114390">some</span> <span m="4114649">constant</span> <span m="4115270">a</span> <span m="4115620">times</span> <span m="4115930">2</span> <span m="4116090">to</span> <span m="4116160">the</span> <span m="4116290">n</span> <span m="4117050">plus</span> <span m="4117240">some</span> <span m="4117420">constant</span> <span m="4117850">b</span> <span m="4118130">times</span> <span m="4118420">3 to the</span> <span m="4118620">n.</span> <span m="4120279">Plug</span> <span m="4120890">it</span> <span m="4120960">in--</span> <span m="4121934">not</span> <span m="4122359">to</span> <span m="4122430">the</span> <span m="4122529">boundary</span> <span m="4122870">conditions.</span> <span m="4123340">Plug</span> <span m="4123609">it</span> <span m="4123660">into</span> <span m="4123840">the</span> <span m="4123920">recurrence</span> <span m="4124380">equation</span> <span m="4125380">and</span> <span m="4125600">solve</span> <span m="4125979">for</span> <span m="4126140">a and b.</span> <span m="4127520">And</span> <span m="4127750">it</span> <span m="4127930">generally</span> <span m="4128270">will</span> <span m="4128350">work.</span> </p>
<p><span m="4130939">If</span> <span m="4131200">g</span> <span m="4131439">is</span> <span m="4131590">polynomial,</span> <span m="4138569">you</span> <span m="4138740">should</span> <span m="4139020">guess</span> <span m="4139279">a</span> <span m="4139330">polynomial</span> <span m="4140450">of</span> <span m="4140569">the</span> <span m="4140680">same</span> <span m="4140970">degree.</span> <span m="4147840">So</span> <span m="4148029">for</span> <span m="4148200">example,</span> <span m="4155810">say</span> <span m="4156069">g</span> <span m="4156380">of</span> <span m="4156460">n</span> <span m="4157930">is</span> <span m="4158220">n</span> <span m="4158439">squared</span> <span m="4159020">minus</span> <span m="4159410">1.</span> <span m="4161109">You</span> <span m="4161229">should</span> <span m="4161479">guess--</span> <span m="4167622">let's</span> <span m="4168069">see--</span> <span m="4170130">guess</span> <span m="4171399">an</span> <span m="4171890">squared</span> <span m="4173050">plus</span> <span m="4173689">bn</span> <span m="4174779">plus</span> <span m="4175080">c.</span> <span m="4176229">Plug</span> <span m="4176880">that</span> <span m="4177060">in</span> <span m="4177210">for</span> <span m="4177380">f</span> <span m="4177490">of</span> <span m="4177620">n.</span> <span m="4179046">All right?</span> <span m="4179450">So</span> <span m="4179580">when</span> <span m="4179670">we're</span> <span m="4179790">doing</span> <span m="4179960">these</span> <span m="4180170">guesses,</span> <span m="4180779">we're</span> <span m="4181200">guessing</span> <span m="4181505">f of</span> <span m="4181810">n.</span> <span m="4184186">Same</span> <span m="4184680">thing up</span> <span m="4185174">here.</span> </p>
<p><span m="4207930">Now,</span> <span m="4208520">say</span> <span m="4208640">you</span> <span m="4208800">mixed</span> <span m="4209130">two</span> <span m="4209410">together.</span> <span m="4210800">Suppose</span> <span m="4211280">you</span> <span m="4211370">have</span> <span m="4211610">an</span> <span m="4211690">example</span> <span m="4212200">where</span> <span m="4213430">g</span> <span m="4213680">of</span> <span m="4213770">n</span> <span m="4215130">equals</span> <span m="4215630">2</span> <span m="4215680">to</span> <span m="4215970">the</span> <span m="4216150">n</span> <span m="4216670">plus</span> <span m="4216960">n.</span> <span m="4218710">What do you</span> <span m="4219010">suppose</span> <span m="4219160">you do in  that</span> <span m="4219617">case?</span> </p>
<p><span m="4222359">AUDIENCE: [INAUDIBLE]</span> <span m="4222816">to</span> <span m="4223273">guess</span> <span m="4223730">[INAUDIBLE].</span> </p>
<p><span m="4224190">PROFESSOR: Yeah,</span> <span m="4224610">guess</span> <span m="4224890">each</span> <span m="4225080">one</span> <span m="4225330">separately,</span> <span m="4225790">add</span> <span m="4225940">them</span> <span m="4226040">together.</span> <span m="4227660">So</span> <span m="4227840">you're</span> <span m="4227970">going</span> <span m="4228080">to</span> <span m="4228140">guess</span> <span m="4230870">f</span> <span m="4231070">of</span> <span m="4231180">n</span> <span m="4232300">equals</span> <span m="4232730">a</span> <span m="4233070">times</span> <span m="4233390">2</span> <span m="4233550">to</span> <span m="4233610">the</span> <span m="4233750">n</span> <span m="4234370">plus</span> <span m="4234780">bn</span> <span m="4235480">plus</span> <span m="4235960">c.</span> <span m="4237530">All</span> <span m="4237960">right?</span> <span m="4238110">Because</span> <span m="4238370">you</span> <span m="4238460">take</span> <span m="4238680">the</span> <span m="4238710">guess</span> <span m="4239010">for</span> <span m="4239080">that</span> <span m="4239460">plus the</span> <span m="4239919">guess for that.</span> </p>
<p><span m="4244050">All</span> <span m="4244340">right.</span> <span m="4244560">Now,</span> <span m="4244660">there's</span> <span m="4244840">one</span> <span m="4245410">last</span> <span m="4246010">thing</span> <span m="4246300">that</span> <span m="4246390">can</span> <span m="4246690">go</span> <span m="4246890">wrong</span> <span m="4247210">here.</span> <span m="4248540">And</span> <span m="4248590">that</span> <span m="4248790">is</span> <span m="4249090">you</span> <span m="4249210">can</span> <span m="4249350">try</span> <span m="4249610">your</span> <span m="4249760">guess,</span> <span m="4251090">and</span> <span m="4251260">it</span> <span m="4251480">doesn't</span> <span m="4251760">work.</span> <span m="4253890">All</span> <span m="4254000">right?</span> <span m="4254340">So</span> <span m="4254570">there's</span> <span m="4254780">rules</span> <span m="4255140">for</span> <span m="4255220">that</span> <span m="4255500">too.</span> <span m="4258460">For</span> <span m="4258590">example,</span> <span m="4260960">if,</span> <span m="4262970">say,</span> <span m="4263300">g(n)</span> <span m="4263920">is</span> <span m="4264090">2</span> <span m="4264240">to</span> <span m="4264320">the</span> <span m="4264480">n</span> <span m="4266140">and</span> <span m="4268080">your</span> <span m="4268230">guess</span> <span m="4270968">of</span> <span m="4271440">a</span> <span m="4271720">times</span> <span m="4272120">2</span> <span m="4272270">to</span> <span m="4272320">the n,</span> <span m="4272570">where  a is</span> <span m="4272870">a</span> <span m="4272930">constant,</span> <span m="4273440">fails.</span> <span m="4274575">And</span> <span m="4274920">we'll</span> <span m="4275160">do</span> <span m="4275500">an</span> <span m="4275560">example</span> <span m="4276000">in a</span> <span m="4276080">minute where it</span> <span m="4276420">fails.</span> <span m="4277800">What</span> <span m="4277980">you</span> <span m="4278060">do</span> <span m="4278180">then</span> <span m="4278600">is</span> <span m="4278750">you</span> <span m="4279000">guess</span> <span m="4280530">a</span> <span m="4281140">polynomial</span> <span m="4281830">times</span> <span m="4282220">2</span> <span m="4282370">to</span> <span m="4282420">the</span> <span m="4282580">n.</span> <span m="4282900">So you'd</span> <span m="4283260">guess</span> <span m="4284190">an</span> <span m="4284780">plus</span> <span m="4285110">b</span> <span m="4285860">times</span> <span m="4286420">2 to</span> <span m="4286560">the n.</span> </p>
<p><span m="4288630">And</span> <span m="4288810">if</span> <span m="4288980">that</span> <span m="4289320">fails,</span> <span m="4291630">your</span> <span m="4291740">next</span> <span m="4292080">guess</span> <span m="4292370">would</span> <span m="4292480">be</span> <span m="4292700">an</span> <span m="4293160">squared</span> <span m="4294110">plus</span> <span m="4294440">bn</span> <span m="4295380">plus</span> <span m="4295760">c</span> <span m="4296440">times</span> <span m="4296730">2</span> <span m="4296890">to</span> <span m="4296950">the</span> <span m="4297100">n.</span> <span m="4299130">And</span> <span m="4299280">it</span> <span m="4299370">won't</span> <span m="4299590">happen,</span> <span m="4300080">but</span> <span m="4300270">if</span> <span m="4300610">that</span> <span m="4300900">failed,</span> <span m="4301670">you'd</span> <span m="4301890">guess</span> <span m="4302210">a</span> <span m="4302310">cubic</span> <span m="4302970">times</span> <span m="4303290">2 to</span> <span m="4303500">the</span> <span m="4303690">n.</span> <span m="4303820">You</span> <span m="4303910">keep</span> <span m="4304220">pounding</span> <span m="4304690">it</span> <span m="4304770">with</span> <span m="4305420">another</span> <span m="4305730">factor</span> <span m="4306220">of n in</span> <span m="4306510">front</span> <span m="4307480">if</span> <span m="4307660">the</span> <span m="4307750">guess</span> <span m="4308050">fails.</span> <span m="4310030">And</span> <span m="4310200">that's</span> <span m="4310410">true</span> <span m="4310730">for</span> <span m="4311266">anything</span> <span m="4311980">at</span> <span m="4312090">all</span> <span m="4312260">you would</span> <span m="4312430">be</span> <span m="4312580">doing</span> <span m="4312910">like</span> <span m="4313140">this,</span> <span m="4313830">and</span> <span m="4313940">that'll</span> <span m="4314200">work.</span> </p>
<p><span m="4315810">And</span> <span m="4315980">I don't</span> <span m="4316000">think I've</span> <span m="4316180">ever</span> <span m="4316620">encountered</span> <span m="4316660">an</span> <span m="4317010">example</span> <span m="4317450">where you</span> <span m="4317550">have</span> <span m="4317670">to</span> <span m="4317760">go</span> <span m="4317910">very</span> <span m="4318160">far</span> <span m="4319440">to</span> <span m="4319560">make</span> <span m="4319740">that</span> <span m="4320380">fly.</span> <span m="4324180">Same</span> <span m="4324460">thing</span> <span m="4324590">we</span> <span m="4324690">had</span> <span m="4324960">for</span> <span m="4325080">repeated</span> <span m="4325450">roots</span> <span m="4326260">in</span> <span m="4326540">the</span> <span m="4326690">characteristic</span> <span m="4327170">equation.</span> <span m="4329310">Multiply</span> <span m="4329610">those</span> <span m="4329800">factors</span> <span m="4330140">of n</span> <span m="4330900">in</span> <span m="4331060">front</span> <span m="4331300">of</span> <span m="4331390">it</span> <span m="4331610">to</span> <span m="4331690">get</span> <span m="4331870">to</span> <span m="4332585">the</span> <span m="4332930">answer.</span> </p>
<p><span m="4333845">All right.</span> <span m="4334310">Let's</span> <span m="4334560">do</span> <span m="4334730">one</span> <span m="4335500">more</span> <span m="4335780">example</span> <span m="4336430">where</span> <span m="4336640">it</span> <span m="4336730">fails,</span> <span m="4337210">so</span> <span m="4337310">we</span> <span m="4337380">get</span> <span m="4337850">to see</span> <span m="4338341">what happens.</span> <span m="4359970">Let's</span> <span m="4360190">try</span> <span m="4360560">this</span> <span m="4360770">recurrence--</span> <span m="4362830">f</span> <span m="4363050">of</span> <span m="4363190">n</span> <span m="4364640">is</span> <span m="4364960">2</span> <span m="4365590">f</span> <span m="4365820">of</span> <span m="4365950">n</span> <span m="4366090">minus</span> <span m="4366490">1</span> <span m="4368470">plus</span> <span m="4369260">2</span> <span m="4369450">to</span> <span m="4369510">the</span> <span m="4369760">n.</span> <span m="4371400">And</span> <span m="4371680">the</span> <span m="4371740">boundary</span> <span m="4372110">condition</span> <span m="4372540">is</span> <span m="4372850">f</span> <span m="4373050">of</span> <span m="4373150">0</span> <span m="4374340">is</span> <span m="4374490">1.</span> </p>
<p><span m="4376870">What's</span> <span m="4377140">the</span> <span m="4377240">first</span> <span m="4377480">thing</span> <span m="4377640">I</span> <span m="4377700">do</span> <span m="4380090">for this?</span> </p>
<p><span m="4382090">AUDIENCE: Set it</span> <span m="4382590">equal to</span> <span m="4383090">0.</span> </p>
<p><span m="4384450">PROFESSOR: Set</span> <span m="4384730">it</span> <span m="4384820">equal</span> <span m="4385070">to</span> <span m="4385150">0</span> <span m="4385760">and</span> <span m="4385920">solve</span> <span m="4386240">it.</span> <span m="4386515">Get</span> <span m="4386790">the</span> <span m="4386850">homogeneous</span> <span m="4387550">solution.</span> <span m="4393240">OK.</span> <span m="4393820">So</span> <span m="4395090">what's</span> <span m="4395370">the</span> <span m="4395660">characteristic</span> <span m="4396340">polynomial?</span> <span m="4400290">Alpha</span> <span m="4400630">minus</span> <span m="4401020">2</span> <span m="4402710">is</span> <span m="4402870">0.</span> <span m="4404520">What's</span> <span m="4404740">my</span> <span m="4404890">root?</span> <span m="4406640">2.</span> <span m="4409790">And</span> <span m="4410030">therefore,</span> <span m="4412150">my</span> <span m="4412360">homogeneous</span> <span m="4412990">solution</span> <span m="4413550">is</span> <span m="4414650">c1</span> <span m="4415120">times</span> <span m="4415380">2</span> <span m="4415540">to</span> <span m="4415590">the</span> <span m="4415730">n.</span> <span m="4416460">Well,</span> <span m="4416790">that's</span> <span m="4416900">pretty</span> <span m="4417110">simple.</span> </p>
<p><span m="4418900">What's the next</span> <span m="4419396">step?</span> <span m="4423860">What do</span> <span m="4423940">I have</span> <span m="4424010">to</span> <span m="4424170">find</span> <span m="4424440">next?</span> </p>
<p><span m="4426366">AUDIENCE: Particular.</span> </p>
<p><span m="4426858">PROFESSOR: Particular</span> <span m="4427842">solution.</span> <span m="4432762">All right.</span> <span m="4433270">So</span> <span m="4434100">what</span> <span m="4434320">am</span> <span m="4434400">I going</span> <span m="4434530">to</span> <span m="4434790">guess?</span> <span m="4440210">a</span> <span m="4440500">times</span> <span m="4440760">2 to</span> <span m="4441040">the</span> <span m="4441210">n.</span> <span m="4444260">All</span> <span m="4444410">right?</span> <span m="4445340">So</span> <span m="4445680">let's</span> <span m="4445940">plug</span> <span m="4446390">that</span> <span m="4446620">in,</span> <span m="4447270">a</span> <span m="4447460">times</span> <span m="4447760">2 to the</span> <span m="4447950">n.</span> <span m="4448110">I'm</span> <span m="4448220">going</span> <span m="4448310">to</span> <span m="4448380">plug</span> <span m="4448850">it</span> <span m="4448910">in to</span> <span m="4449140">here.</span> <span m="4450720">So</span> <span m="4450760">I</span> <span m="4450870">get</span> <span m="4451020">a</span> <span m="4451620">2</span> <span m="4451880">to</span> <span m="4451930">the</span> <span m="4452110">n</span> <span m="4452940">equals</span> <span m="4453520">2</span> <span m="4454506">times</span> <span m="4455000">a</span> <span m="4457040">times</span> <span m="4457790">2</span> <span m="4458100">to</span> <span m="4458190">the</span> <span m="4458420">n</span> <span m="4458570">minus</span> <span m="4458970">1</span> <span m="4460610">plus</span> <span m="4461110">2</span> <span m="4461280">to</span> <span m="4461360">the</span> <span m="4461520">n.</span> <span m="4463200">Did</span> <span m="4463310">I do</span> <span m="4463440">that</span> <span m="4463690">right?</span> <span m="4464760">Think</span> <span m="4465180">so.</span> <span m="4467190">So</span> <span m="4468140">I</span> <span m="4468240">get</span> <span m="4468480">2 to</span> <span m="4468680">the</span> <span m="4468800">n,</span> <span m="4469300">2 to</span> <span m="4469470">the</span> <span m="4469540">n,</span> <span m="4469810">2 to the n. I</span> <span m="4470000">get</span> <span m="4470400">a</span> <span m="4472180">equals</span> <span m="4472810">a</span> <span m="4473160">plus</span> <span m="4473520">1.</span> <span m="4476040">Not</span> <span m="4476480">so</span> <span m="4476550">good.</span> <span m="4478510">All</span> <span m="4478620">right?</span> <span m="4479650">There's</span> <span m="4479800">no</span> <span m="4480170">solution</span> <span m="4480700">for</span> <span m="4480860">a.</span> <span m="4482586">Well,</span> <span m="4483070">that's</span> <span m="4483350">bad.</span> </p>
<p><span m="4484220">All</span> <span m="4484410">right.</span> <span m="4484680">So</span> <span m="4485370">what</span> <span m="4485540">do</span> <span m="4485650">I</span> <span m="4485740">do?</span> <span m="4492940">Any</span> <span m="4493120">thoughts</span> <span m="4493460">about</span> <span m="4493630">what I'm  going to do</span> <span m="4494103">next?</span> </p>
<p><span m="4497414">AUDIENCE: [INAUDIBLE].</span> </p>
<p><span m="4498360">PROFESSOR: What</span> <span m="4498500">is</span> <span m="4498620">it?</span> </p>
<p><span m="4498940">AUDIENCE: Change</span> <span m="4499373">your guess.</span> </p>
<p><span m="4500240">PROFESSOR: Change the</span> <span m="4500610">guess.</span> <span m="4501200">What's</span> <span m="4501400">the</span> <span m="4501500">next</span> <span m="4501750">guess going to</span> <span m="4502130">be?</span> <span m="4506780">Yeah,</span> <span m="4507240">all  right.</span> <span m="4507580">So now</span> <span m="4507750">I'm going to</span> <span m="4508210">guess</span> <span m="4511540">f</span> <span m="4511650">of</span> <span m="4511800">n</span> <span m="4513140">is</span> <span m="4513960">an</span> <span m="4514830">plus</span> <span m="4515220">b</span> <span m="4515870">2</span> <span m="4516350">to</span> <span m="4516590">the</span> <span m="4516720">n.</span> <span m="4518630">We'll</span> <span m="4518820">hope</span> <span m="4519030">for</span> <span m="4519160">better</span> <span m="4519430">luck.</span> <span m="4519840">So</span> <span m="4521460">let's</span> <span m="4521660">plug</span> <span m="4521900">that</span> <span m="4522110">in.</span> <span m="4523020">an</span> <span m="4523480">plus</span> <span m="4523790">b</span> <span m="4525850">2 to</span> <span m="4526230">the</span> <span m="4526420">n</span> <span m="4528000">into--</span> <span m="4529020">where am</span> <span m="4529300">I,</span> <span m="4530450">up</span> <span m="4530600">there--</span> <span m="4532100">equals</span> <span m="4533000">2</span> <span m="4533990">a</span> <span m="4534730">n</span> <span m="4534900">minus</span> <span m="4535170">1</span> <span m="4536090">plus</span> <span m="4536460">b</span> <span m="4538290">times</span> <span m="4538980">2</span> <span m="4539230">to</span> <span m="4539290">the</span> <span m="4539460">n</span> <span m="4539580">minus</span> <span m="4539950">1</span> <span m="4541620">plus</span> <span m="4541940">2 to the</span> <span m="4542160">n.</span> </p>
<p><span m="4544435">All</span> <span m="4544890">right.</span> <span m="4545130">Now,</span> <span m="4545260">I</span> <span m="4545330">can</span> <span m="4545470">divide</span> <span m="4545860">out</span> <span m="4546120">the 2 to</span> <span m="4546560">the</span> <span m="4546730">n's</span> <span m="4547010">here</span> <span m="4547480">like</span> <span m="4547740">I</span> <span m="4547840">did</span> <span m="4548010">before.</span> <span m="4551000">And</span> <span m="4551130">I</span> <span m="4551400">get</span> <span m="4554200">an</span> <span m="4554940">plus</span> <span m="4555280">b</span> <span m="4556480">equals</span> <span m="4558070">an</span> <span m="4558820">minus</span> <span m="4559360">a</span> <span m="4560210">plus</span> <span m="4560890">b</span> <span m="4561460">plus</span> <span m="4561930">1.</span> <span m="4564510">That</span> <span m="4564690">cancels</span> <span m="4565240">here,</span> <span m="4566620">b</span> <span m="4566960">cancels.</span> <span m="4569240">So</span> <span m="4569820">I</span> <span m="4569900">got</span> <span m="4570050">a</span> <span m="4570100">solution--</span> <span m="4571750">a</span> <span m="4571950">equals</span> <span m="4572350">1.</span> <span m="4573511">And</span> <span m="4573900">I</span> <span m="4573980">don't</span> <span m="4574280">care</span> <span m="4574640">what b</span> <span m="4574820">is.</span> <span m="4575500">I</span> <span m="4575780">didn't</span> <span m="4575960">need</span> <span m="4576010">to</span> <span m="4576270">set</span> <span m="4576350">it to</span> <span m="4576840">anything.</span> <span m="4577910">So</span> <span m="4578030">I'll</span> <span m="4578210">make</span> <span m="4578400">it</span> <span m="4578490">0</span> <span m="4578930">just</span> <span m="4579160">to</span> <span m="4579440">get it</span> <span m="4579670">out</span> <span m="4579820">of</span> <span m="4579870">the</span> <span m="4579970">way</span> <span m="4580880">because</span> <span m="4581070">it</span> <span m="4581160">didn't</span> <span m="4581360">matter.</span> </p>
<p><span m="4582960">So</span> <span m="4583110">my</span> <span m="4583250">particular</span> <span m="4583800">solution</span> <span m="4584310">then</span> <span m="4585760">is</span> <span m="4588090">f</span> <span m="4588310">of</span> <span m="4588440">n--</span> <span m="4590320">a</span> <span m="4590790">is</span> <span m="4590960">1.</span> <span m="4591375">b</span> <span m="4591790">I</span> <span m="4591840">can</span> <span m="4591990">just</span> <span m="4592180">make</span> <span m="4592510">0</span> <span m="4592820">because</span> <span m="4592980">it</span> <span m="4593270">didn't</span> <span m="4593460">matter</span> <span m="4593700">what</span> <span m="4593920">I</span> <span m="4594000">used,</span> <span m="4594370">so I'll</span> <span m="4594440">make</span> <span m="4594620">it</span> <span m="4594730">simpler.</span> <span m="4595790">So</span> <span m="4595910">I</span> <span m="4595990">get</span> <span m="4596220">n</span> <span m="4596670">2</span> <span m="4596990">to</span> <span m="4597220">the</span> <span m="4597380">n.</span> <span m="4598670">That's my</span> <span m="4598910">particular</span> <span m="4599397">solution.</span> <span m="4602810">And</span> <span m="4602980">I've</span> <span m="4603080">got</span> <span m="4603320">my</span> <span m="4603840">general</span> <span m="4604250">solution</span> <span m="4604720">up</span> <span m="4604840">there</span> <span m="4605060">as</span> <span m="4605160">c1</span> <span m="4605770">times</span> <span m="4606030">2</span> <span m="4606190">to</span> <span m="4606240">the</span> <span m="4606400">n--</span> <span m="4607750">sorry,</span> <span m="4607800">the</span> <span m="4607870">homogeneous</span> <span m="4608420">solution.</span> </p>
<p><span m="4611530">What's</span> <span m="4611880">the</span> <span m="4612690">next</span> <span m="4612850">step?</span> <span m="4619207">What's</span> <span m="4619700">step</span> <span m="4620100">3?</span> <span m="4624570">What</span> <span m="4624700">do</span> <span m="4624810">I</span> <span m="4624900">do</span> <span m="4625110">with</span> <span m="4625210">these</span> <span m="4625420">guys,</span> <span m="4627530">this</span> <span m="4627720">solution and</span> <span m="4628190">that</span> <span m="4628390">solution?</span> </p>
<p><span m="4629387">[? AUDIENCE: Add them. ?]</span> </p>
<p><span m="4630580">PROFESSOR: Add</span> <span m="4630850">them</span> <span m="4630990">together.</span> <span m="4631740">Good.</span> <span m="4632060">So</span> <span m="4632130">the</span> <span m="4632230">general</span> <span m="4632580">solution</span> <span m="4638640">is</span> <span m="4638970">I</span> <span m="4639080">have</span> <span m="4640070">f</span> <span m="4640330">of</span> <span m="4640430">n</span> <span m="4640850">is</span> <span m="4641050">the</span> <span m="4641150">sum</span> <span m="4642200">of</span> <span m="4642730">c1</span> <span m="4643055">2</span> <span m="4643380">to</span> <span m="4643630">the</span> <span m="4643790">n</span> <span m="4644600">plus</span> <span m="4644950">the</span> <span m="4645040">particular</span> <span m="4645480">solutions</span> <span m="4646000">n</span> <span m="4646310">2 to</span> <span m="4646520">the</span> <span m="4646680">n.</span> <span m="4647700">How</span> <span m="4647870">do</span> <span m="4647970">I</span> <span m="4648050">figure out what</span> <span m="4648240">c1</span> <span m="4648738">is?</span> </p>
<p><span m="4651228">AUDIENCE: Plugging  in</span> <span m="4651726">to</span> <span m="4652224">[INAUDIBLE].</span> </p>
<p><span m="4652730">PROFESSOR: Plugging</span> <span m="4653220">in</span> <span m="4653780">the</span> <span m="4654620">boundary</span> <span m="4655140">condition--</span> <span m="4656460">f(0)</span> <span m="4657050">equals</span> <span m="4657320">1.</span> <span m="4658275">f(0)</span> <span m="4659440">equals</span> <span m="4659800">1.</span> <span m="4660400">Plug</span> <span m="4660735">in</span> <span m="4661070">0,</span> <span m="4661500">I</span> <span m="4661580">get</span> <span m="4662650">c1</span> <span m="4662900">2</span> <span m="4663140">to</span> <span m="4663370">the</span> <span m="4663470">0</span> <span m="4664840">plus</span> <span m="4665850">0.</span> <span m="4667240">Well,</span> <span m="4667670">that's</span> <span m="4667830">pretty</span> <span m="4668030">easy.</span> <span m="4668290">c1</span> <span m="4668480">equals</span> <span m="4668680">1.</span> <span m="4671390">So</span> <span m="4671630">I</span> <span m="4671710">now</span> <span m="4671940">have</span> <span m="4672180">the</span> <span m="4672250">final</span> <span m="4672600">answer.</span> <span m="4674230">f</span> <span m="4674360">of</span> <span m="4674480">n</span> <span m="4675180">equals</span> <span m="4675760">2</span> <span m="4676000">to</span> <span m="4676080">the</span> <span m="4676240">n</span> <span m="4676900">plus</span> <span m="4677390">n</span> <span m="4677760">2</span> <span m="4678130">to</span> <span m="4678210">the</span> <span m="4678430">n.</span> <span m="4679940">All right?</span> <span m="4680865">Any questions?</span> <span m="4685820">OK.</span> <span m="4686230">So</span> <span m="4687070">it's</span> <span m="4687250">a</span> <span m="4687740">little</span> <span m="4688000">tedious</span> <span m="4689200">to</span> <span m="4689310">do</span> <span m="4689460">this.</span> <span m="4689740">But</span> <span m="4689890">the</span> <span m="4689970">really</span> <span m="4690190">nice</span> <span m="4690480">thing</span> <span m="4690750">is</span> <span m="4691410">any</span> <span m="4691800">linear</span> <span m="4692140">recurrence</span> <span m="4692560">you</span> <span m="4692660">ever</span> <span m="4692870">see,</span> <span m="4694000">this</span> <span m="4694230">method</span> <span m="4695100">always</span> <span m="4695470">works,</span> <span m="4696110">which</span> <span m="4696360">is</span> <span m="4696480">handy.</span> <span m="4697590">OK.</span> <span m="4697890">That's it</span> <span m="4698190">for</span> <span m="4698490">today.</span> </p>
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